The Problem of Stipulation

I think it would surprise people to read Engelmann and Carnine’s Theory of Instruction. (The previous link is to the 2016 edition of the book on Amazon, but you can find the same book for free here.) Tangled noticeably within its characteristic atomism and obsession with the naïve learner are a number of ideas that seem downright progressive—a label never ever attached to Engelmann or his work. One of these ideas in particular is worth mentioning—what the authors call the problem of stipulation.

problem of stipulation

The diagram at left shows a teaching sequence described in the book, in all its banal, robotic glory.

To be fair, it’s much harder to roll your eyes at it (or at least it should be) when you consider the audience for whom it is intended—usually special education students.

Anyway, the sequence features a table and a chalkboard eraser in various positions relative to the table. And the intention is to teach the concept of suspended, allowing learners to infer the meaning of the concept while simultaneously preventing them from learning misrules.

Stipulation occurs when the learner is repeatedly shown a limited range of positive variation. If the presentation shows suspended only with respect to an eraser and table, the learner may conclude that the concept suspended is limited to the eraser and the table.

You may know about the problem of stipulation in mathematics education as the (very real) problem of math zombies (or maybe Einstellung)—which is, to my mind anyway, the sine qua non of anti-explanation pedagogical reform efforts.

But of course prescribing doses of teacher silence is only one way to deal with the symptoms of stipulation. Engelmann and Carnine have another.

To counteract stipulation, additional examples must follow the initial-teaching sequence. Following the learner’s successful performance with the sequence that teaches slanted, for instance, the learner would be shown that slanted applies to hills, streets, floors, walls, and other objects. Following the presentation of suspended, the learner would be systematically exposed to a variety of suspended objects.

Stipulation of the type that occurs in initial-teaching sequences is not serious if additional examples are presented immediately after the initial teaching sequence has been presented. The longer the learner deals only with the examples shown in the original setup, the greater the probability that the learner will learn the stipulation.

It’s a shame, I think, that more educators are not exposed to this work in college. It’s a shame, too, that Engelmann’s own commercial work has come to represent the theory in full flower—unjustly, I believe. Theory of Instruction could be much more than what it is sold to be, to antagonists and protagonists alike.

Update (07.13): It’s worth mentioning that, insofar as the problem of stipulation can be characterized as a student’s dependence on teacher input for thinking, complexity and confusion can be even more successful at creating cognitive dependence than monotony and hyper-simplicity. When students—or adults—are in a constant state of confusion, they may learn, incidentally, that the world, either at the moment or in general, is unpredictable and incommensurate with their own attempts at understanding. In such cases, even the unfounded opinions of authority figures will provide relief.

Audio Postscript


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Josh Fisher

Instructional designer and editor for K-12 mathematics. My research interests center mostly around mathematics education.

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