If you ask the prototypical adult who took AP Calculus in high school what a “derivative” is, you’ll generally get a half-decent answer, assuming they were reasonably engaged in class. You might hear that it’s a slope, a rate of change, a measurement of how fast something moves. You’ll get something actionable. If you ask the same adult to compute an elementary derivative in front of you, they’ll almost certainly fail, even if they aren’t too far removed from their time in class. This is not a failure of their education by any means. In some ways, it is the only meaningful success that said education currently has.
Curricula can attempt to instill two things in a student. The first is procedural fluency: the ability to perform the steps of some algorithm on demand, to execute a technique, to perform a computation. We tend to think of this as the most important. But the second thing is more durable, even if less outwardly crisp: an epistemic residue, a working sense of what the concept being learned about fundamentally is, what kinds of claims it can support and what would refute it, and where it lies within the bigger picture of knowledge that the student has accumulated over time. In the United States, curricula in mathematics, the sciences, and the limited curricula that exist in computation are designed and assessed essentially exclusively against the first. This is in direct contrast to what survives in the graduates of these curricula ten, twenty, fifty years later — almost exclusively the second thing, delivered almost entirely by serendipitous accident, in the gaps between the procedures the Curriculum was actually intended to teach.
Concepts and epistemic residues persist over time, while procedures without regular reactivation do not. The cognitive infrastructure that our Curricula provide adults, the backbone by which they are intended to lead their lives and function in an increasingly technological society, bears almost no resemblance to the infrastructure which the curriculum was ostensibly optimized for. Worse, this asymmetric forgetting is not bidirectional in its effects. An adult who retained the concept can quickly re-acquire the related procedures on demand. The adult who, cramming for their examinations, once learned the procedure but knew nothing of the concept will be unable to run the reverse. There is nothing here that the procedure could have left behind, for the scaffolding to which it naturally should’ve attached was never built. Our curricula optimize for the thing that doesn’t last, at the chief expense of the thing that does, and the thing that does is the thing from which all else follows.
The Mechanism
The asymmetrical forgetting rests on a distinction older than the cognitive science vocabulary widely adopted to describe it. Procedural knowledge and conceptual knowledge are not stored, retrieved, nor reactivated in the same way, and they consequently do not decay in the same way.
Procedural knowledge is inherently sequential. To compute a derivative is to execute a series of moves in a particular order, conditioned only on the form of the input. To balance a chemical equation, to write a for-loop in a syntax that has not been recently used, or to manually run long division for the first time since fourth grade — these are all chains of steps whose only durable representation in the brain is the chain itself. Such chains inherently require reactivation to persist. The adult who has not balanced an equation in fifteen yearsaa Or, sadly, the recent college graduate who hasn’t done it in a mere six years… embarrassing, I know! has not forgotten because they were incapable or because they were poorly taught; they have forgotten because the sequential chain hasn’t fired in fifteen years, and chains that are not run inevitably decay. This is not controversial. It is the easy direction, the half of the asymmetry that the curriculum tacitly acknowledges through myriad practice problems and lifeless examinations. What the curriculum fails to acknowledge is the expiration date: the day the student stops being a student.
Conceptual knowledge is structured fundamentally differently. A concept is not a sequence but a schema. This allows for integration into the rest of what one understands about the world they inhabit. The adult who has retained the basic concept of the derivative has retained it because the schema gets reactivated incidentally, for the ordinary course of living provides suitable analoguesbb I would go further to hypothesize that any reasonable schema will with probability ~1 be reactivated, for everything is correlated.. The schema is constantly reactivated; every news article mentioning acceleration, every casual remark about curves of stocks becoming steeper, every passing thought about how quickly something is growing. The procedural skill has no such ambient reactivation. There is nothing in adult civic life that can incidentally re-run the steps of the quotient rule. The schema persists by virtue of the world’s analogues consistently reaching in and touching it; the procedure decays because it is left devoid of interaction that the curriculum once served to forcibly provide.
The final component of this mechanic is the most severe. Consider two adults, one who has retained the concept and lost the procedure, and the other who has improbably (but for the sake of argument) retained the procedure without ever having the concept. We will continue with our example of differentiation. The first adult, on encountering a problem that requires a derivative, will find the procedure from a reference, say a web search, and re-acquire it in minutes. The concept’s scaffolding has built a place to store the procedure when it returns. The second adult, faced with an identical problem, cannot recognize that it requires a derivative in the first place, for they never had such a scaffolding; the schema that would enable them to notice is markedly absent. Even if the procedure is entirely intact, it has nowhere to attach to, no occasion on which to be deployed. A procedure cannot magically summon a concept that was never built.
From this final component follows the weight of the asymmetry as a design constraint. If both of these aspects of cognitive infrastructure were equally durable, or even equally recoverable in reduction, the question of which to prioritize would be a matter of taste, pedagogical convenience, and moderate pretentiousness. This is, of course, not the case. The conceptual component is the substrate within which the procedural becomes meaningful, the only component surviving long enough to matter for the adult life the retired student will lead. A curriculum that optimizes for students who can blindly execute procedures they will lose in five years has produced essentially nothing of lasting value. It has produced nothing more than a cacophonous credential that lacks any semblance of underlying understanding.
Substantiation
Where do I land amidst all of this, and why do I care? I offer my own retention audit of sorts not as proof of the mechanism, but as a demonstration that it is at least observable in lived form. If nothing else, perhaps you, dear reader, can run such an audit on yourself and see if the results are the same.
I attended a rural public school in upstate New York. By every metric that the system has conceived, I was a success by the time of my graduation. I was in the top ten of my class, I had straight As on my state examinations in mathematics and the sciences, I had all 4s and 5s on my AP examinations (including some which were not even offered by my institution), and graduated with the highest honors possible in my district, heading outbound to an Ivy League university. The audit forces reframing: of what was taught to me with procedural intent, what has survived, and in what form?
What do I remember of stoichiometry? I got a perfect 100 on my chemistry examination in high school, converting between grams and molecules, running limiting-reagent problems. I was clearly good at it at the time, and yet I remember nothing of how to balance even a trivial equation without re-deriving it slowly from first principles. I have not computed a mole quantity since my examination in 2020. What I retain from my Chemistry experience, other than the fact that providing troublemaking high school students with Bunsen Burners is outright objectionable at best, is that chemical reactions are quantitative, that matter is conserved across them, and the relationships between reactants and products are precise. I retained all of the concepts despite the curriculum such that when I went on to take advanced courses in mathematics and physics at Brown, the scaffolding was laid; I could connect what I learned to form a bigger picture. The procedure has decayed because I have not balanced an equation since I was sixteen, but the structural concept has survived.
What of AP Biology, another course of which I earned a perfect score on the final examination? I will not bore the readers with another long-winded description. The procedures that I once mastered, say those for Punnett Squares, have long left me. Yet the concepts are strong enough that in the years since, I have been able to do research work that is strictly integrated with medicine and the life sciences. When I need a procedure from these fields that are outside of my expertise, I have the conceptual scaffolding to place what I derive into.
The pattern is identical across every subject that I can examine. What was taught with mere procedural intent has decayed, while what has survived is the epistemic and conceptual residues, the schemas. The success of my education, by my own retention audit, is a success that the curriculum was not optimizing for. It is not, therefore, a success that can be attributed to some quality of that curriculum or quality of the public institution at which I studied. It is the pattern of a curriculum that failed at what it tried to do and, accidentally, in the failure, left behind for me the only thing of value.
I do not believe that my case is unusual, and I invite you to perform such an audit on yourself if you feel open to it. Ask yourself: of any procedural unit that you remember being drilled on for exams, what survives now? Is it the procedure that was assessed or the concept that was incidentally surfaced alongside it? The mechanism predicts what your audit will find, and by virtue of my results, I’d place my money on the same prediction.
Implication
So far we have been concerned with the individual. The individual graduate, twenty years on, retains the concept and loses the procedure by failure of the curriculum. At this scale, the consequence is little more than regrettable in a bounded way. The individual is poorer for the loss, and perhaps there has been some time squandered away by the system, but the promise of the residue may remain, providing enough to build on if they ever so choose to put in a bit of effort. One can see why we might just shrug our shoulders and keep on walking right past this, for people will muddle through, and the people who do care can refresh and derive for themselves.
The true consequence of this fact lives at the scale of the population. The population that emerges from the widespread adoption and delivery of such a curriculum is the population that has to operate the society all graduates inhabit. Let us consider, then, what the mechanism predicts at such a scale. The median adult, twenty years on, has neither working procedural fluency nor the robust conceptual scaffolding that would enable subsequent procedural acquisition. The shards of residue that they do retain are little more than incidental, surfaced in the gaps between the procedures that the curriculum tried to teach, never deliberately developed, never assessed, and never built into the structure that the curriculum optimized for. It is thin where it should be thick, accidental where it should be load bearing.
This population is the one that we then ask to do the work that civic life requires. We ask this population to evaluate claims made by scientific institutions during a pandemic. We ask them to vote on the regulation of technologies they have never been taught to reason about, to navigate algorithmic and financial systems whose underlying structure and principles they have no schema for, to distinguish a credible statistical claim from a contrived and misleading one. The asymmetric forgetting mechanism predicts, and with high accuracy, that the population we have is widely unable to do these tasks at the level that is implicitly required. This is not due to incapability, for even a thin conceptual residue would provide meaningful opportunities, but it is rather because the curriculum is optimized for the wrong part of what it could instill and leave behind in the long term. We live inside the aggregate consequence of this fact.
This is made worse by the fact that it compounds. Each generation of teachers is drawn from a population produced by the curriculum of the previous generation. My rural hometown teachers had themselves passed through a curriculum optimized for procedural fluency. Whatever conceptual residue they retained was that from which their teaching stemmed. The conceptual depth that asymmetric forgetting calls for simply cannot be requested of teachers who were themselves never taught to that depth. It is not a reasonable expectation, and is thus not a personal failure of the teachers. The current teacher workforce that cannot provide a concept-first curriculum is rather the predictable product of a curriculum that did not build concepts deeply. Those who are interested in the concepts must find it within themselves to search further in the current setting. The path from where we are to a system that optimizes for the concepts runs through the teachers and their own educations, and that path is inevitably long. There is no magic solution that will resolve this in six months, nor within a glorious five year plan. Recognizing this is not a counsel of despair, but rather a counsel of patience, a refusal to mistake or conflate the difficulty of the path for evidence against the destination.
I have deliberately chosen not to describe what a concept-first curriculum would look like. This is a separate piece of work that is owed its own diligent treatment, and I refuse to attempt to distill and collapse it into the closing of this one. What follows from the asymmetry is the subject of work to be done and essays yet to be written, by myself and by others.
Coda
The adult who remembers what a derivative is at the conceptual level has been given something. The curriculum that gave it to them did so by accident, in the margins between what it was actually optimized to deliver, and at the cost of everything else that it could’ve deliberately built into the same student. The residue is real and is genuinely the only thing that survived. It is also catastrophically less than what twelve years of schooling could have left behind, if only the system had known what it was for.
The asymmetry is not subtle and it is not new. We have been running this experiment on every cohort of American students for as long as “American students” have existed. The results have now been replicated millions of times. Those results are the population that we now have. A population that retains the wrong half of what it was taught as a thin accident, subsequently tasked with operating a society whose questions and demands require the half that is absent. The curriculum may have optimized for what would not last, but what actually lasted is the accidental byproduct. We ask that byproduct to do the work, but the work is too large for such an accident to bear.
We can choose different. The asymmetry shows us exactly what to reach for: the epistemic residue that persists, the schemas that the world’s analogues continually reach in to refresh, and the conceptual scaffolding to which future learning can attach. We are choosing against this. We have been choosing against it for a very long time. The cost of that choice is borne not by the student who took the exams but by the adults they became, the society those adults now have to navigate without the infrastructure their schooling could have built.
The most valuable thing a curriculum can give a student is what the student will still have twenty years down the line. We are giving them everything else.