It’s Thursday evening and you got a 63 on the linear algebra quiz. You know what you’re going to do about it: redo the whole problem set. All twelve problems, start to finish. Maybe do it twice. Put in the time, fix the score.
Here’s what’s almost certainly going to happen: you’ll do problems one through five quickly and competently, because you understood those. You’ll slow down on six and seven, muddle through. You’ll get stuck on eight — the same kind of problem that killed you on the quiz — and eventually look at the answer. You’ll finish the set feeling like you covered the material.
Next quiz: 65.
The problem isn’t effort. It’s how you distributed it. You spent most of your practice time on things you already know how to do, and you spent just enough time on the hard part to get through it without actually cracking it open.
Practice is not the same as improvement
Most studying is repetition, not improvement. You go through problems, pages, flashcards. The rep count goes up. The effort feels real. But repetition and improvement are not the same thing, and confusing them is one of the most expensive mistakes you can make with limited time.
K. Anders Ericsson, the psychologist who spent decades studying how expertise develops, made this distinction the center of his research. In his framework of deliberate practice, real improvement requires: identifying a specific failure point, isolating it, practicing it in focused repetition, and getting feedback on whether it improved. What most people do instead is what Ericsson called “naive practice” — doing the same thing repeatedly and hoping familiarity accumulates into skill.
The tennis player who hits balls for two hours is practicing. The tennis player who identifies that their second serve breaks down under pressure, sets up a simulation of exactly that scenario, and hits fifty second serves under increasing pressure conditions — that player is doing deliberate practice. The two-hour hitter might improve slightly from sheer exposure. The targeted player improves faster per hour by a significant margin.
In Ericsson’s foundational 1993 paper with Krampe and Tesch-Römer, published in Psychological Review, they found that what distinguished elite musicians from good musicians wasn’t total hours — it was the proportion of practice time spent in deliberate, effortful, specifically targeted improvement work. The elite performers didn’t just practice more. They practiced differently.
The failure point is the point. Not the whole skill. Not the whole problem set. The exact joint where the execution breaks down.
Finding the break point — and drilling only that
Here’s what drilling the failure point actually looks like in academic study.
You got problem eight wrong on the quiz. Not just kind of wrong — you got it wrong in a particular way. Maybe you set up the matrix correctly but made an error in row reduction. Maybe you got the row reduction right but didn’t know what to do with the result. Maybe you got completely lost at step one.
Each of those is a different failure point and requires a different drill.
If you set up the problem correctly and broke down in row reduction, you don’t need to redo the whole problem. You need to drill row reduction in isolation: take five different matrices, do only the row reduction step, check your work, do it again. The setup isn’t the weak link. The setup will take care of itself once row reduction is solid.
If you couldn’t get past step one, the issue is upstream — probably in conceptual understanding, not procedure. That’s a different kind of drill: close the notes, try to write down in plain language what problem eight is actually asking, then look at your notes to see where your description diverged.
The point is to be surgical. The problem set as a whole is the map. Your wrong answer is the X. You don’t need to walk the whole map. You need to go to the X.
Tonight’s session: ten minutes on the hardest step
Here’s what to do with this tonight.
Find a problem from class you got wrong this week. Not the hardest problem overall — the one where you got the most stuck, or where the answer you produced was furthest from correct. That’s your failure point.
Before you redo the problem, do this: identify exactly which step broke down. Write it down. “I couldn’t factor this expression.” “I got the formula right but made an algebra error in the second line.” “I didn’t know which formula to apply.” That diagnosis is where you start, not “I got problem eight wrong.”
Now set a 10-minute timer. During those ten minutes, drill only that step — in five different variations. If you couldn’t factor the expression, don’t redo problem eight. Find four more expressions and factor them. Then find a fifth. Set up the step, execute it, check it, note where it went wrong, repeat.
The variations are important. If you drill the same failure in the same context, you learn to handle that specific context. If you drill it in five different contexts, you learn the underlying move.
Ten minutes. One failure point. Five variations. That’s the whole protocol.
After the ten minutes, do one complete version of a similar problem to test whether the step is fixed in context. If it is, you’re done with that failure point for tonight. If it isn’t, you’ve at least identified a more specific breakdown to drill tomorrow.
The thing that makes this hard is that it requires you to accurately diagnose your failure point before you start drilling. That diagnosis is itself a skill, and students who aren’t used to it will tend to be vague: “I just don’t get matrices.” That’s not a failure point, that’s a category. Push further: within matrices, which step. Within that step, which part of the step. The more precisely you can name what broke, the more precisely you can fix it.
Why this works better than the whole problem set
The reason doing twelve problems again produces marginal improvement is that problems one through five are essentially rest. They’re confirmation that you know things you already knew. They might feel good, but they’re not where the improvement lives.
The improvement only lives in the vicinity of the failure. Ericsson’s research consistently showed that deliberate practice is uncomfortable in a specific way: you’re spending almost all your time near the edge of your current ability, where failure is common. That’s not a design flaw. That’s the design. The edge is where the growth is.
This doesn’t mean you should torture yourself with only the hardest material every session. There’s a role for review and consolidation. But when you have a specific test coming and a specific gap that showed up last week, spending two hours on what you already know is not the same investment as spending thirty minutes on what you don’t.
The phrasing that sticks: “Don’t practice until you get it right. Practice until you can’t get it wrong.” The second standard requires specifically targeting the failure until it’s gone, not reviewing everything until it all feels familiar.
If you’re regularly identifying failure points but forgetting them between sessions, the active recall method that makes retrieval stick pairs directly with this — retrieval tells you what the failure point is, deliberate practice is how you close it. And if you find yourself spending the ten minutes before your timer starts watching your phone instead of diagnosing the failure, the procrastination loop has a specific fix that doesn’t require willpower.
You already know what broke. Drill that. Leave the rest alone.