I have a tendency, when writing blog posts, to leave important things unsaid. So, let me fix that up front before I forget. What I wanted to say here was that, in my view, learning doesn’t happen unless we tick off all three boxes: encoding, consolidation, and retrieval.

It’s not that learning gets **better or stronger** when more of those boxes are ticked off. Learning isn’t **possible**—to some degree of certainty—in the first place without all three. And it’s not the case that focusing on just retrieval instead of just encoding or just consolidation represents some kind of revolution in pedagogical thinking. You’re simply ignoring one or two vital components of learning when before you were ignoring one or two other ones. Learning can still happen even when we don’t think about one or two (or all three) of the above components, but then it’s haphazard, random, implicit, and/or incidental. (In that case, learning comes down to mental horsepower and genes rather than processes over which we have some control.) All three components still must be addressed for learning to occur; it’s just that we can decide to not be in control of one or all of them (to students’ detriment).

But even then we’re not done. We’ve covered the components of the process of learning, but all three of those components intersect with another dimension of learning, which describes the products of learning: thinking about and thinking with.

Thinking With

In the previous post linked above, the examples of slope could all be categorized as “thinking about” slope. Put too simply, encoding the concept of slope means absorbing information about slope, retrieving knowledge about slope means remembering the slope concept and saying your knowledge out loud or writing it down, and consolidating knowledge about slope means practicing, such that what is encoded stays encoded and what is known can be retrieved.

All of this—encoding, consolidation, and retrieval—must happen with “thinking with” as well as with “thinking about.” Encoding–Thinking With, for example, would involve absorbing information about how slope can be applied to do other things, whether mathematically or in the real world. Common examples include designing wheelchair ramps (which have ADA-recommended height-length ratios of 1 : 12), measuring and comparing the steepnesses of things, and determining whether two lines are parallel or perpendicular (or not either). Consolidating–Thinking With would involve practice with that encoded knowledge—solving word problems is a typical example. Finally, Retrieving–Thinking With would involve remembering that encoded knowledge, particularly after some time has passed, say by using slope to solve a programming problem or a problem on a test.

All six boxes have to be checked off for learning to occur (such that it is within our control).

Teach Thinking With

In education, we have difficulties—again, in my view—with Encoding–Thinking With, and Consolidating–Thinking With. As far as these two are concerned, it is rare in my experience to see guidance and practice on a wide variety of different problems involving thinking with (for example) slope to answer questions that aren’t about slope. We tend to think that all we should do is give students a bunch of think-about slope facts and then hope for them to magically retrieve and apply those to thinking-with situations. We misunderstand transfer as some kind of conjuring out of thin air, so that’s what we give students—thin air. Then we stand back and hope to see some conjuring. When this doesn’t produce results that we’d like, we—for some unimaginably stupid reason—blame knowing facts for the problem, and instead of supplementing that knowledge with thinking-with teaching, we **swap them**. Which is much worse than what it tries to replace.

Instead, it is necessary to **teach** students how to think with slope and many other mathematical concepts, and to provide them with practice in thinking with these concepts. Thinking with is as much knowledge as thinking about is.

One of my favorite examples of thinking with slope—and one which I have, admittedly, not yet written a lesson about—has to do with drawing convex or concave quadrilaterals. Given 4 coordinate pairs for points that **can** form a convex or concave quadrilateral (no three points lie on the same line, etc.), how can I decide, somewhat algorithmically, on the order in which I should connect the points, such that the line segments I actually draw do create a quadrilateral and not the image on the far right?

One way to go about it is to first select the leftmost point—the point with the lowest x-coordinate (there could be two, with equal x-coordinates, but I’ll leave that to the reader to figure out). Then calculate the slope of each line connecting the leftmost point with each other point. The order in which the points can be connected is the order of the slopes from least to greatest. This process would create a different proper quadrilateral than the one shown in the middle above.

Checking Off All the Boxes

Students’ minds are not magical. They don’t turn raw facts magically into applied understanding (the extreme traditionalist view), and they don’t magically vacuum up knowledge hidden in applied contexts (the extreme constructivist view). Put more accurately, this kind of magic does happen (which is why we believe in it), but it happens outside of our control, as a result of genetic and socioeconomic differences, so we can take no credit for it.

Importantly, ignoring components of students’ learning, for whatever reason, subjects them to a roll of the dice. Those students who start behind stay behind, and those who are underserved stay so. We seem to have enough leftover energy to try our hand at amateur psychology and social-emotional learning. Why not take a fraction of that energy and channel it into, you know, plain ol’ teaching?