Prerequisite chains

Why can two missed weeks of mathematics haunt a student for years when the same absence in history costs little?

A child is off ill for a fortnight in March. In history she misses the Tudors, reads about them over Easter, and by June nobody could tell. In mathematics she misses fractions, and three years later she is still coming apart on rates, ratio and algebraic manipulation. Same absence, same child, same fortnight. The usual explanation is that mathematics is harder or more abstract — but difficulty is a property of a topic, and what we are trying to explain is a property of time. Why does one gap heal and the other compound?

Start by drawing the two subjects as graphs rather than lists. Ask of any two topics: does the second consume the output of the first? In history, the Tudors and the French Revolution are siblings. They share ancestors — reading, chronology, weighing a source — but neither is an input to the other, so the graph fans out from a small common trunk. In mathematics, equivalent fractions feed ratio, which feeds proportional reasoning, which feeds linear functions and rates of change. That is a chain.

Now the arithmetic of the two shapes, because it is the whole argument. Suppose a topic is learned properly with some probability p when everything it depends on is in place. In a fan, missing one node costs you one node: losses add. In a chain of length k, arriving intact at the far end requires every link to hold, so the chance is p multiplied by itself k times. Losses compound. That single difference — adding versus multiplying — is enough to turn an equal-sized initial gap into two completely different futures, without either subject being harder than the other.

But a probability is a bookkeeping device, not a mechanism. What actually goes wrong at the next link? Not simply ignorance — the child who half-knows fractions can still, with effort, work one out. The cost is where the effort goes. Fluent recall costs almost nothing and leaves working memory free for the new procedure; reconstructing the sub-step consumes the very capacity the new topic needed. So she does not merely lack the prerequisite — she learns the dependent topic worse than her classmates while apparently doing the same lesson. That is how one hole manufactures the next one.

There is a second, quieter mechanism, and I think it does more damage. Consider when the gap becomes visible. In history it announces itself as itself: I don't know about the Tudors. It is legible, local, and fixed by reading. In mathematics the fractions gap is silent in March and surfaces in November as failure at ratio — it presents wearing the wrong name. So the help she receives is aimed at ratio, more practice at the thing built on the hole, which is close to the least useful intervention available.

And the class does not wait. Curriculum pace is set for the group, so repair must be bought on top of the current week's work rather than instead of it. A gap in a chain therefore accrues something like interest, repaid from a budget already fully spent. That is what makes this path dependence rather than a one-off loss.

Let me try to falsify this. If the mechanism is graph shape rather than subject identity, the pattern should cut across subjects, not along them. Within a language, missing the introduction of a tense system should haunt like fractions — and it does. Within mathematics, the more encyclopaedic material — names of solids, statistical vocabulary — should heal like the Tudors, and it does. Meanwhile the chain-like spine of history, a working sense of what came before what, does behave cumulatively. The account predicts within-subject variation and gets it, which is better evidence than the between-subject contrast we started from.

Two refinements. First, this is a claim about dependency, not difficulty. A hard topic with no dependants is expensive once; an easy topic with many dependants is expensive indefinitely. Sequencing, not rigour, makes a subject unforgiving.

Second, mathematics anxiety is real and often offered as the explanation here. I would put it downstream: repeated failure at topics built on a hidden gap is an excellent way to manufacture the belief that one cannot do mathematics, and that belief then suppresses the practice needed to repair the gap. It is a genuine amplifier and probably a partial cause, but on its own it does not explain why the damage is delayed by months and appears at a specific later topic.

The honest weak point is the evidence, so let me be plain about it. Absence from school is not random — children who miss a fortnight differ in many ways from those who do not — so observational studies of absence cannot cleanly separate the gap from the child. The nearest thing to an exogenous break is the long summer holiday, and older meta-analytic work put the loss at roughly a month of grade-equivalent progress, larger in computation than in reading, which is the direction this account predicts. But that literature is contested: later reanalyses argue much of the apparent loss depends on how tests are scaled across grades, and some datasets show little or none. Treat the direction as reasonably supported and any specific magnitude as unsettled — it is the softest number here, which is why I have given no other.

• How to diagnose backwards from a failing topic to the actual missing prerequisite.
• Whether mastery-based sequencing, which refuses to advance until a link holds, trades the compounding problem for a pacing one.

Every term the collection defines is gathered in the glossary.