I think the main idea of my previous post is that academic learning is essentially a set of tricks, all the way down. Because many of these tricks are sped up and automatized, and learning them has faded in memory, we can’t notice all their tricky parts in consciousness—we are practically forced to think about them as higher-level constructs, because that is all we can see when we look at our own thinking and problem solving.

We see this, for a somewhat strained example, when looking at our thinking, which we would describe (if you can see it) as a circle going back and forth from left to right in a loop.

But that's only a description of what you **see** happening. It is not, in fact, what **is** happening at all. And I don't just mean that the computer knows nothing about "circle," "back and forth," or "left to right," although that's true. I mean that, even at the higher level of the code itself, what is happening is that the canvas is being completely erased, the circle center is moved left or right 4 pixels according to a RULE, and then the circle is redrawn at the new location.

This is easy to swallow when the end result is just a circle going back and forth on the screen. But it's important to remember that scaling up the complexity of this end result is not an indication that the process that produced the circle changed in any fundamental way. The most thrilling thing you saw in a theater this year operates under the same basic principle.

Although I need to move on to the research I want to talk about in this post, I think the implications of the above (if they are basically correct) deserve a lot more reflection in education. Certainly one implication worth pondering is that anyone who is successful at a certain endeavor is, by reason of this success alone, no better than an amateur at describing the replicable mechanism for that success.

"Here's the numbers I used to win the lottery" –Entrepreneurs giving advice

— Andrew Wilkinson (@awilkinson) July 25, 2017

Yet, for endeavors related to academic learning at least, this does not mean that replicable mechanisms don't exist.

The Magic of Solving Problems

In mathematics education, problem solving is the bouncing ball, the entrepreneurs' advice. The processes that **seem** to govern its successful execution in any particular case are almost certainly not the ones that actually do. As a demonstration of this, Darch, Carnine, and Gersten (1984) investigated an explicit instruction method for teaching fourth graders how to translate mathematics word problems involving division and multiplication into equation form. This was tested against what the authors called "basal instruction," which they describe, in part, as follows:

The teacher-guided portion of the lesson involved two components: (a) discussion designed to increase student involvement and motivation; and (b) teacher presentation of strategies to solve problems. . . .

The second key element in the teacher-guided segment was presentation of an approach for developing problem-solving strategies. This aspect appeared in each of the four state-adopted texts . . . A four-step system was presented to the students with the steps sequentially placed in a line above boxed areas for writing. The steps were: (1) place numbers here; (2) identify correct operation; (3) write and complete number sentence; and (4) place your answer here.

You'll note that although the instruction here provides a system for attacking problem solving in steps, it does not teach students **how** to identify the correct operation in Step 2. Carnine and Gersten's explicit instruction method, described—again, in part—below, focused almost exclusively on this step.

Students first learned to discriminate multiplication problems from addition problems. The rule students were taught was: "If you use the same number again and again, you multiply." . . .

The exercises introduced the students to the concept of "number families," and the relationships between the "big number" and two "little numbers" in each family (for multiplication and division) . . . students also learned that when the big number was not given, they multiplied the two smaller numbers to determine it. Conversely, when the big number was given, the students divided. . . .

In the final step of the teaching sequence, students learned to ask two questions to discriminate among addition, subtraction, multiplication, and division story problems: Does the story deal with the same number again and again? [if so, it's multiplication or division] . . . Does the story give the big number? [if so, it's multiplication; if not, division].

Results

As you might have guessed while muttering something about teaching to the test like I did, students taught using the explicit method outperformed those taught with the typical "basal instruction." On a 26-item posttest containing multiplication, division, addition, and subtraction word problems, which assessed only students' "ability to write the correct computation statement," and not correct answers, students taught using the explicit strategy obtained an average score of 86.5%, while their counterparts taught with typical basal instruction scored, on average, 63.7%. Thirty-one of the 36 students in the explicit-strategy group scored at or above 80%. In the basal group, just 9 students, out of 38, did the same.

Why Do We React So Negatively to This Type of Instruction?

A typical reaction to reading the details of explicit strategies like the above involves two mistakes, I think. The easier mistake is to assume that the method is supposed to represent the full flower of childrens' work with understanding and solving word problems and problem solving in general. We are simply too used to the idea that modern pedagogies must represent their intended end result at every scale of interaction with students, and, conversely, we must take every level of interaction with students—from activity to course—as representative of a pedagogy's intended aim. The harder mistake is to assume that you can increase the generality of instruction sorely missing from the explicit method outlined above by making it less direct or less explicit. It is certainly a mistake to think that if you have taught it students have learned it. But it's something close to light-speed stupid to assume that this implies that if you **don't** teach it, they **will** learn it.

Regardless, though, a big problem with this negativity (from me, included) is that it prevents us from learning and improving our own instruction. The method presented in this paper reminds me of my days working with elementary-level texts and fact triangles. Using fact triangles side by side with operational problem solving as a kind of model is a really cool idea. I don't have to use the terms "big number" and "little numbers." And I don't have to imagine at any moment that I'm giving up and just teaching these kids how to get good scores on the end-of-year test.

What I have to face is that there is something about problem solving that I can't see when looking at my own thinking, and what I can't see that I'm doing **might very well be something like "Does the story deal with the same number again and again? [if so, it's multiplication or division] Does the story give the big number? [if so, it's multiplication; if not, division].**

Reference: Craig Darch, Doug Carnine and Russell Gersten, *The Journal of Educational Research*, Vol. 77, No. 6 (Jul–Aug., 1984), pp. 351–359