Regularities, All the Way Down

Kris Boulton, Engelmann and the hard last mile of instructional design

In 2019, I sat in the audience at researchED London to watch Kris Boulton deliver a talk titled “Cognitive science is almost useless for designing effective teaching.” To a room full of cognitive science enthusiasts, it was the equivalent of shouting “Fire!” in a crowded theatre.

The reaction was one of general bewilderment. Part of the problem was rhetorical; Kris, much like his intellectual lodestar Siegfried Engelmann, tends to use many words to describe things most people would prefer were said in few. To make matters worse, the discussion was derailed by an irrelevant side remark that annoyed a prominent scientist in the room.

But underneath the friction, Kris was trying to argue that cognitive scientists give us theories of learning, but not theories of instruction. They tell us how the brain processes information, but they don’t tell us what to say to a room of thirty 14-year-olds on a Tuesday morning.

At the time, I didn’t have the right conceptual hooks to make sense of the importance of how Kris had come to view instruction. Now I can see that the distinction between theories of learning and those of instruction isn’t actually interesting to me. (After all, it’s possible that if we really did fully understand cognition then instruction would become more self-evident.) Instead, I’ve come to see Kris’ struggles with instructional design as one of navigation of different “layers” of regularity.

Layers of regularities, and the temptation to stop too soon

In a previous post, I wrote about regularities in learning: not iron laws, but stable patterns that can be described well enough to design around. I also suggested that these regularities form a hierarchy.

1. The universal layer of general cognitive science

Cognitive science mostly trades in the general laws of the human animal that apply to almost anything worth learning. This is the “biology” of education. It tells us that working memory is limited and that retrieval strengthens long-term memory. It provides the boundary conditions for instruction, but it doesn’t provide a script.

2. The knowledge type layer

One level down, we find patterns that recur within families of knowledge or content classes. This is where Siegfried Engelmann operated. He proposed that if we can correctly classify knowledge into “teachable forms”—such as Basic Concepts, Joinings, or Cognitive Routines—then the method of instruction begins to dictate itself.

3. The individual idea layer

Finally, there are the regularities of the individual idea itself. These are the things that must be attended to if you are to teach an idea successfully. What matters here is getting awkward definitions right, knowing the persistent “near-neighbour” confusions, or representation quirks. It is what Lee Shulman famously called Pedagogical Content Knowledge (PCK).

It is extraordinarily tempting to believe that the higher layers will ultimately make the lower ones unnecessary. There are good reasons for this temptation. One is intellectual: human beings are pattern-hungry. We want the clean theory that explains the messy particulars. We want the rule that makes the exceptions disappear. Another is practical: a world in which each tiny topic requires its own hard-fought instructional discovery is a deeply inconvenient world. It would be labour-intensive for teachers, and it would be labour-intensive for machines. If the “right way” to teach limits, or negative numbers, or ratios is genuinely idiosyncratic — if it cannot be recovered from general principles or content classes — then scaling excellent instruction looks less like deploying a recipe and more like building a map, one small landmark at a time.

Why Kris Boulton is a Useful Case Study

This is where Kris’ journey becomes interesting — and, frankly, why I think more people should be paying attention to him. He is one of the smartest living thinkers about mathematics instruction, not because he has a novel ‘framework’, but because he is willing to do the slow work of making Engelmann’s framework bite.

Kris was one of a handful of UK educators who felt the shortcomings of cognitive science early on and became convinced that Engelmann held the missing structure: a way to specify instruction tightly enough that failure could be made rare and intelligible, rather than normal and vaguely blamed on children. But because Engelmann hadn’t developed a suitable higher-level mathematics course, Kris needed to begin the work of applying Engelmann’s principles to England’s secondary mathematics curriculum. How difficult could it be?

Very difficult, it turns out. If you can find a dozen or so spare hours to wade through Kris’ interviews on Craig Barton’s podcast or follow his more recent writing on Substack, you can hear how his thinking has unfolded over time. It’s a story of someone discovering what you only discover when you stop theorising at the level of principles and try to build something properly.

In his earlier work designing instruction for the online platform UpLearn, and now in building Unstoppable Learning to train teachers in “instructional engineering”, the story has been a hard-won realisation. Kris has done an extraordinary amount to make Engelmann’s idea of “content classes” effective in secondary mathematics. Yet even when you classify a topic correctly, you are not finished: the content class constrains what good instruction can look like, but it does not tell you everything you need to know to teach the idea well.

If you are teaching limits, Engelmann’s theory of instruction will tell you what content class “limit” belongs to and it will tell you some principles for working out how to teach it, but you have to do the hard thinking yourself. Do you treat “limit” as a prediction, a tolerance, or a repair? Each route seeds a different long-run understanding in the minds of students. Similarly, should you first introduce imaginary numbers as a formal extension that makes algebra “complete”, or geometrically, or through formal symbolism that postpones deeper interpretation?

The difficulty is not that Engelmann’s theory is wrong, but rather that “content classes” constrains instruction, but do not uniquely determine it. This is because there are decisive regularities that appear to be local:

• What misconception is most likely?

• What first representation should carry the meaning?

• Which easy shortcut must be prevented because it produces brittle learning later?

• Which distinction must be made explicit immediately because it will otherwise remain invisible until it causes a collapse several topics later?

So what should we conclude?

We often hear that education theory has “failed” because it cannot tell teachers exactly what to do. But the more accurate conclusion is that theory often operates too high up the hierarchy and then over-promises.

Cognitive science does give us genuine regularities, and they matter. But they do not tell us how to teach any particular idea. Engelmann’s contribution was to push one level down: to propose that there are regularities in what knowledge is like that tell us something about how instruction should proceed. That is an enormous advance.

And yet, as Kris’ journey to try and apply Engelmann to secondary mathematics shows, there is a further layer beneath that: idea-level design. If the most valuable knowledge lives at the level of “how to teach this specific idea,” then teacher development needs to be less like high-level exhortations (“be clear,” “use retrieval”) and more like the slow building of a shared library: worked solutions to recurring design problems, made explicit enough to be reused. This is the work that Kris is currently undertaking with teachers in mathematics across multiple countries, and it is interesting to watch his progress from a distance.

This insight should also sober anyone building learning platforms. Platforms do not succeed through technological development alone; they succeed when they encode the structural features that make particular ideas hard—misleading intuitions, fragile distinctions, and the most reliable entry routes. LLMs can help considerably here, because they can draw on a vast archive of teacherly explanation and instructional writing. But that only matters if platform designers recognise that this idea-level guidance must be made explicit and built into the instructional design, rather than left to emerge by accident.

The good news is that this work can be done. The bad news is that it looks less like discovering a single grand theory or pipeline, and more like building a map—one difficult idea at a time.

---

If you care about the craft of instruction — in any subject — Kris is worth following. Even when the examples are mathematical, the underlying problem is universal: how you turn an idea and a framework into something that can be reliably learnt by anyone.