A student does a multiplication problem and gets the wrong answer. The teacher marks it wrong and shows them the correct method. The student practices more. Next test: wrong again.
What happened? The teacher diagnosed a practice problem and prescribed more practice. But the student's actual problem wasn't that they needed more practice. Their actual problem was a specific wrong belief about how multiplication works — one that produced the wrong answer consistently and systematically, and that no amount of correct-method demonstration was going to fix.
Getting more practice at the correct method when you have a wrong belief at the foundation is like building on a cracked foundation. The more you build, the bigger the eventual collapse.
The question isn't just what they got wrong. It's why they got it wrong. The why determines everything about what to do next.
An execution error is when the student understands the concept but made a mistake in the doing — an arithmetic slip, a notation error, a transcription mistake, running out of time. If you ask them to explain what they did, they'll describe a correct process. They often catch the error themselves while explaining. The fix is practice — more exposure to the operation so the execution becomes more reliable.
A schema error is when the student has a wrong belief about the concept itself. Their reasoning chain is consistent — it's just built on an incorrect premise. If you ask them to explain what they did, they'll describe a process that contains a specific wrong step. They believe the explanation is correct. The fix is not more practice — it's addressing the wrong belief first, before any further instruction.
After a wrong answer, ask: "Walk me through what you did."
If the student describes a correct reasoning chain and the error is a slip — that's an execution error. Tell them their reasoning is correct, note the slip, and move on. Practice will fix it.
If the student describes a reasoning chain that contains a wrong premise — that's a schema error. Stop. Name the wrong premise specifically. Do not show them the correct method yet. Fix the belief first. The method follows from the correct belief.
If the student can't explain what they did at all — that's absent knowledge. Instruction, not correction.
If you keep getting the same type of problem wrong despite practicing — stop practicing the same thing. You may have a wrong belief, not a practice deficit. The way to find out: try to explain to yourself why the method you're using should work. If your explanation is wrong somewhere, that's the problem. Fix the explanation. The execution follows.
If you can explain your method correctly and you still get the answer wrong — that's an execution problem. Practice at that operation specifically, timed if possible, until it becomes automatic.