What a good explanation has to do

An explanation can sound brilliant. I can follow every sentence and still get stuck when I try to use the idea. Asking AI for another version is easy. I want the teaching to find out why I'm stuck.

In How I approach learning, I described teaching that responds to what I know and directs effort toward understanding. Here is what I want that to look like, one explanation at a time.

A small example lets us inspect the reasoning closely. I also want the explanation to help me approach situations where the assumptions haven't already been supplied. What should I pay attention to? What would I need to find out before using this idea?

Find the reasoning behind the answer

Consider a bag containing three red counters and two blue counters. Each counter is equally likely to be drawn. Draw twice without putting the first counter back. What is the probability of drawing red both times?

Suppose a learner answers 9/25. That answer alone leaves several possibilities open. They might have overlooked the instruction about replacement, assumed that the second draw has the same probabilities as the first, or applied a familiar multiplication rule without checking its conditions.

The next useful question might be: after the first red counter is removed, what is left in the bag? That brings the reasoning into view. If they describe two red counters among four, we have a different problem to investigate than if they still describe three among five.

This is what I want from an initial assessment. Ask me to attempt something and explain the attempt, then use it to decide what to address. My confidence can add context, but the tutor still needs to inspect the reasoning. A few answers give us a provisional starting point that can change during the lesson.

Make the destination clear

Before explaining the calculation, I want to know what the explanation is trying to achieve. Here, the target is to understand how removing a counter changes the possibilities for the next draw, and why that changes the calculation.

That gives us a way to judge progress. Can I identify the assumption? Explain what follows from it? Say what would change if the assumption changed?

In How People Learn, the National Research Council describes expert knowledge as including the conditions in which it is useful. Knowing a principle and knowing when to use it are connected parts of understanding. I want the explanation to make those conditions visible, including when the method would be inappropriate. (2000, Chapter 2, pp. 43–44)

Formative assessment uses judgments about work to guide improvement. In Sadler's theoretical account, learners need an appreciation of the quality being aimed for, a way to compare their work with it, and ways to improve. His focus is on qualitative judgments using multiple criteria. I would apply that idea to the quality of my explanation: its assumptions, the steps connecting them, and the conclusion they support.

The route should make those connections visible. For this example, we might need equally likely outcomes, fractions of a set, and probability conditional on an earlier event. If one of those ideas is missing, we can work on it. If I already use it comfortably, the explanation can build from there.

Give me something to think about

When I have enough context, I want a small question to engage with before the next explanation. After drawing a red counter and keeping it out, is the chance of red on the next draw above, below, or equal to 3/5? Why?

Now the explanation has something specific to address. Two red counters remain among four, so the chance of red on the second draw, given that the first was red, is 2/4. The chance of two reds is 3/5 × 2/4 = 3/10. Each fraction refers to a particular situation; the second depends on what happened in the first.

The calculation gives me another way to inspect my prediction. If I expected no change, I need to account for the counter that has left the bag.

This resembles pretesting: attempting a question before studying its answer. In experiments comparing testing before and after study, the relative benefits depended on the assessment; the pretesting advantage in the delayed experiment was confined to repeated questions. That supports considering a brief attempt, while leaving room to teach first when I cannot yet make sense of the question.

The point of my prediction is to expose an assumption we can examine. I want an accurate explanation afterward and a chance to revise what I thought.

Give enough help to make reasoning possible

Sometimes the question itself is beyond my starting point. Repeating it or telling me to think harder gives me no new way forward. A worked example supplies a solution whose steps I can inspect. Scaffolding is support that can be reduced as I become more capable.

The amount of guidance matters. In a statistics instruction experiment, learning activities interacted with prior knowledge on immediate conceptual understanding. Lower-knowledge learners benefited from worked examples compared with generating arguments. The study supports adapting guidance; it supplies no universal threshold for withdrawing it.

For the counter problem, the help could be a drawing of the bag after the first red is removed. If I understand that picture but cannot connect the fractions, we can examine that connection. Another explanation could label the individual counters and count the equally likely ordered draws. There are five choices for the first draw and four for the second. Six of those twenty pairs contain two red counters, giving the same 3/10.

These representations give me ways to inspect the reasoning. Once I can follow it, the tutor can leave a step for me to supply, then ask me to handle a related case with less support. What I can do should influence the next move.

Let my explanation change what comes next

I want to explain the idea back in my own words. Self-explanation means generating an account of how a step follows or how new information connects with what I already know. Repeating the explanation's wording gives the tutor less to work with than making the relationships explicit.

In the original circulatory-system study, students prompted to explain a text showed greater gains than students who reread it. The prompted group also spent substantially longer studying, and the posttest allowed text and notes. That limits what the comparison can establish about explanation alone.

For my purposes, the explanation gives us something to examine. Suppose I get 3/10 but say I multiplied because the draws are independent. The arithmetic is right; the justification needs repair. In this example, the probability of red on the second draw changes with the first outcome. The multiplication uses that conditional probability.

Feedback should help me locate and repair the mismatch, then give me a chance to try again. If the AI challenges a step, I want the reason for the challenge. We can check the calculation, compare it with a dependable source, or inspect another case. The tutor's confidence does not settle whether its correction is sound.

Check before moving on

A lesson can feel smooth while leaving important work undone. In a randomized introductory-physics study, students doing active work performed better on immediate tests while reporting less perceived learning than students hearing fluent lectures. That distinction gives me a reason to check what I can do, while telling me little about whether any particular frustrating session is useful.

For the bag example, put the first counter back before drawing again. What changes, and why? The second draw now has three red counters among five again, so the probability of two reds is 3/5 × 3/5 = 9/25. Explaining why the two situations differ gives us more useful evidence than repeating the original calculation.

Suppose I want to estimate the chance of two components failing together. Knowing each one's failure probability isn't enough to justify multiplying them: I'd need a reason to treat the failures as independent. A shared power supply, for example, gives me something to investigate. I'd want the teaching to help me identify what information is missing and what to check before choosing a calculation.

Here, explaining the idea means helping me examine the situation. Recognizing that I don't yet have enough information can be a useful outcome.

My answer should affect the lesson. If I can justify the change, we can move on. If I use a correct rule for the wrong reason, we can examine the assumption. If the next question is inaccessible, we can return to a missing prerequisite or show another example.

That is the teaching conversation I want: my reasoning helps determine the explanation, and my next attempt helps determine what happens afterward. Research informs those choices; this particular sequence remains a teaching proposal. A successful check tells us something about that moment. Practice will ask whether I can use the idea later, recognize when it matters, and work through a situation with the sources and tools available to me.