Chapter 13 · Schemas Come from Comparison

If far transfer is blocked because structure cannot become a cue, there is one path: to detach the structure itself from the surface and store it in long-term memory as a single node that can light up, set apart, on its own. A structure thus separated from the surface and settled in as an independent composite representation is called a schema. With a schema established, it is summoned in any situation answering to that structure, whatever the material. But two questions hide in that one sentence. How does that structure node come to be, and once it exists, what lights it again in a new situation? We solve the two in turn.

Comparison Forges the Structure

A structure node scarcely arises however deeply you dig a single case. Look deeply at the fortress story alone, and what remains is a representation bound to army, road, and fortress. The structure "gathering at one point" stays stuck to that concrete material and does not light in the tumor problem, whose surface differs. There is even a paradox: the deeper you sink into one case, the more firmly you are bound to that case's surface.

Detaching structure from the surface happens in comparison. Lay two or more cases with different surfaces side by side in working memory and compare them, and one side's army, road, and fortress and the other's ray, tumor, and body diverge with nowhere overlapping. But while you pair and match the two, there is something that comes to lie in the same place: the relations of dividing-and-sending and gathering-at-one-point take the same role on both sides. The non-overlapping surface nodes cancel one another, and only the relations that lie in the same place, matched, remain. Bind this remaining skeleton of relations into one unit and encode it, and a structure node bound to no surface at last stands. Comparison is the work of aligning—shaving off the two cases' differing flesh against each other and culling only the relations that come to lie in the same place.

So given the same time, comparing two or more cases with different surfaces leaves a more transfer-able representation than digging one case deeply. A learner who dug only one gains a representation bound to that surface and reaches only similar problems; a learner who compared several gains a surface-stripped structure node and summons it even at an unfamiliar surface. The difference between a novice filing a problem under its material—"this is a tank problem," "that is a train problem"—and an expert filing the same problems under their structure—"these are all ratio problems"—comes from here.

What Lights That Node

Setting up the structure node solves only half. Any node is retrieved only when its members light up in working memory, and the structure node's members are not objects like army or ray but relations like "dividing a force into several branches" and "gathering at one point." So meeting the tumor problem and having ray, tumor, and body rise into working memory does not, on its own, light this node. Those surface objects are not the structure node's members, so there is nowhere overlapping. "If the surface differs you cannot reach the structure"—this is the very place that line was about.

Then what is the cue that lights that node? The schema "divide into several branches and gather at one point" is, in the end, also a composite representation binding simpler relations, and its members are "divide" and "gather." Since any node is retrieved only when its members light up in working memory, to light this schema, "divide" and "gather" must float in working memory. So the question narrows to one: in the newly encountered problem, can you raise that "divide" and "gather" into working memory? This is what divides whether the schema lights or not.

But what rises for free on meeting a problem is the surface objects alone. Ray, tumor, and body rise straight off when seen, but neither "divide" nor "gather" is written on that surface. Those two rise into working memory only when you sketch how to solve this problem—what if I split the ray into several beams and gather them at one point? Only processing that goes beyond taking down the surface to building the relation beyond it stands the members up. This processing does not happen for free; it takes working memory.

Only after "divide" and "gather" stand in working memory does activation spread to the schema that has the two as members, and the node lights. The structure node is not laid on from above first to organize the problem. The node lights as a result of the member relations rising from below, and only the node thus lit then reflects the whole problem back as one structure. Organizing is not the cause of retrieval but its result. The cue is always those member relations actually floating in working memory, and the node comes after.

"Divide" and "gather" are relations anyone knows, but expert and novice divide at raising them into the new problem and receiving them as one structure. The divide is twofold. One is whether you have a composite representation binding the two. Without a schema, even if the two relations rise, there is no node to receive them and they merely float apart; with a schema, that weave is retrieved at once as one structure. The other turns on whether you are practiced at the processing of building the relation beyond the surface and raising the members. Raising abstract members into working memory is the same work you had to do when forging the schema by comparison, so the more often you have done it, the more easily you stand the members up in a new problem too. That an expert recognizes structure straight off in an unfamiliar problem is not a special eye but the merit of having a schema to receive it and being practiced at standing the members up.

This finishes telling why the so-desired far transfer is rare. What rises for free in a new problem is always the surface, and the relations that would be members are often hidden on the solution side, not the problem, so they do not even stand in working memory before you reckon how to solve it. If a surface-resembling old problem is readily pulled out and you stop there, "divide" and "gather" never rise and even the schema you have stays asleep. Someone telling you "look at that story just now," or your resolving to reckon the structure yourself, is precisely raising those members into working memory. Failing to raise the members that would light the structure you have, and so passing it by—that is the difficulty of far transfer. Why one who has once reached mastery comes to see structure ever more easily lies in this interlocking.

The Double Edge of Concreteness

From this property of the index come the two faces of the concrete example. In teaching an abstract rule, giving a familiar concrete case is common and natural; even a first-seen rule, laid on a familiar case, has many cues to hook and enters the head easily. But that very richness of surface trips you up. A rule learned through one concrete case alone gets bound to that case's surface nodes and is not summoned in a new problem with a different surface. A learner who mastered fraction division through measuring cups alone, seeing a problem with no measuring cup as something wholly different, is like this. The rule was clearly learned, but it is trapped in the surface of the measuring cup and does not light elsewhere.

Conversely, master it with abstract symbols stripped of concrete flesh, and with little surface to bind to, it is easy to carry across many surfaces. Bound to no material, it reaches that much more widely. But with cues to hook scarce, it is hard to learn at first. With no concrete to set foot on, the abstract floats in the air. Concreteness aids entry but blocks transfer; abstractness aids transfer but blocks entry. They are two sides of the same property of the index.

The diagnosis of a double edge itself tells the direction of the prescription. At the threshold, let them enter on a concrete case, but rather than stay there, have them compare cases with different surfaces side by side so as to detach the structure from the surface. Enter by the concrete and exit by comparison. Spoon-feed one case and end it, and they are trapped in its surface; have them compare several, and they strip the surface off.

One Part We Have Met All Along

This surface-stripped structure node is not something we meet for the first time in this book. Earlier, an expert holding a whole chess game whole while using little of working memory's narrow room was thanks to having bound the scattered individuals into one unit; that bound unit is precisely the surface-stripped structure node. The background knowledge drawn on when filling a text's gaps to build a model of the situation in the head was also a bundle of relations that works whatever the material is changed to—a structure node. The one thing used as a composite representation that saves working memory, as a frame that fills comprehension's gaps, and as transfer's bridge, we have been meeting all along in different places.

This one property—that the index hangs by the surface—bred the difficulty of transfer and set the way schemas, which overcome that difficulty, are formed and operate. But the same property breeds another trouble: when several composite representations hang together on one surface cue and contend over the path. How that contention blurs retrieval and draws in misconceptions—the story of interference is next.