Phil Daro has a line that's worth saying out loud to a class: the right answer is boring. The mistakes are interesting.
He doesn't mean answers don't matter. He's spent years making exactly that distinction: correct answers are essential, but they're part of the process, not the product, and the product is the math students walk away with in their heads (Daro, 2014). What's interesting is the part that builds that math, the thinking. Once a student says "twelve," you know one thing: they said twelve. The part you can actually teach from is how they got there. And that's true for the strongest students as much as the ones who are struggling. A correct answer with no thinking showing is a closed door. The thinking behind it is the whole house.
This is a stance worth handing to students, not just holding as a teacher. We want them to find the thinking interesting, not only to be right. A class that treats "how did you get that?" as the good part is a different kind of class than one racing to call out the answer.
A bare answer teaches no one
Picture two students. One writes the correct answer and nothing else. The other writes a wrong answer with every step of their reasoning visible. Which one can you teach from?
The wrong answer, every time. It tells you exactly where the student's understanding has a gap — which step went sideways, what they believed that turned out not to hold. The bare correct answer tells you nothing you can use. You don't know if they understood it, guessed it, or copied it. The thinking is the part you can work on; the answer is just the part you can score.
That's the pivot under every one of these routines, said plainly: away from did you get it and toward show me how you're thinking. The activity exists to get student thinking out where you and the class can see it, because thinking you can see is thinking you can work on. Thinking that stays in a student's head, behind a correct answer, is invisible to everyone, including sometimes the student.
This is bigger than catching mistakes
It would be easy to file this under "error analysis" and move on. That's too small.
Some of the routines have no wrong answers to catch. Sorting graphs by their features. Ranking four functions by how fast they grow. Finding a defensible reason each of four objects could be the odd one out. There's no error to surface, and the move is exactly the same: away from landing on a result and toward making the reasoning visible. Why did you group those together? What feature were you tracking? What would change your ranking?
So this isn't really about mistakes. Mistakes are one case of it. The whole point is visible thinking, and a wrong answer is just one especially useful kind of visible thinking, because the gap it reveals is so clear.
The research that visible thinking pays off
It's not only that visible thinking is useful to you as the teacher. The act of making thinking visible changes what the student learns.
Students who explain their reasoning as they study a worked example learn more than students who study the same example passively. Michelene Chi and colleagues called this self-explanation, and found the students who generated explanations for themselves understood more and transferred it further (Chi et al., 1989). The explaining wasn't a readout of learning that had already happened. The explaining was where the learning happened.
Errors specifically, when they're surfaced and explained rather than buried, drive stronger learning than clean, error-free performance (Metcalfe, 2017). In algebra, students who studied and explained incorrect worked examples improved their equation-solving more than students who saw only correct ones (Barbieri & Booth, 2020). The wrong example, examined, taught more than the right one, copied.
And comparison — putting two different approaches to the same problem side by side and asking what's the same and what's different — drives both deeper understanding and more flexibility in how students solve (Rittle-Johnson & Star, 2009). But comparison only works if the thinking is visible in the first place. You can't compare two approaches that were never shown.
What this builds in students over time
There's a longer game here than any single lesson.
The National Research Council named productive disposition as one of the strands of real mathematical proficiency: the habit of seeing math as sensible, useful, worth thinking about, and seeing yourself as someone who can do it (Kilpatrick, Swafford & Findell, 2001). That disposition doesn't come from a poster. It comes from what a classroom treats as interesting day after day.
A class where visible thinking is the good part — where the surprising approach gets airtime, where "wait, why did you do that?" is a compliment — is a class where that disposition can grow. An answer-only classroom quietly teaches the opposite: that math is a sorting machine for who's fast and who isn't, and the thinking is just the boring stuff in between getting it right. Which classroom a student spends a year in shapes what they come to believe math is.
The Monday move
Resist the pull to land on the right answer and move on.
When something half-right or surprising shows up, slow down. Ask the student to show the thinking. Put it where the class can see it. The instinct, especially when the clock is tight, is to confirm the answer and keep moving — but that's the moment you skip the only part that teaches. The visible thinking, the half-right, the unexpected, the "why did you go that way," is not a detour from the math. It's the math, made visible long enough for everyone to learn from it.
The right answer really is the boring part. Everything worth teaching is in how it got there.
References
Barbieri, C. A., & Booth, J. L. (2020). Mistakes on display: Incorrect examples refine equation solving and algebraic feature knowledge. Applied Cognitive Psychology, 34(4), 862–878.
Chi, M. T. H., Bassok, M., Lewis, M. W., Reimann, P., & Glaser, R. (1989). Self-explanations: How students study and use examples in learning to solve problems. Cognitive Science, 13(2), 145–182.
Daro, P. (2014). Against "answer-getting" [Video]. SERP Media. https://serpmedia.org/daro-talks/
Kilpatrick, J., Swafford, J., & Findell, B. (Eds.). (2001). Adding it up: Helping children learn mathematics. National Academies Press.
Metcalfe, J. (2017). Learning from errors. Annual Review of Psychology, 68, 465–489.
Rittle-Johnson, B., & Star, J. R. (2009). Compared to what? The effects of different comparisons on conceptual knowledge and procedural flexibility for equation solving. Journal of Educational Psychology, 101(3), 529–544.