What Algebra Tutor actually means
The phrase Algebra Tutor gets used loosely enough that it is worth pinning down before going any further.
Two words, two jobs: Algebra names the approach and Tutor names who or what it is for. Most confusion comes from treating them as one idea.
The reason undergraduates, postgraduates and adult learners returning to study end up here is rarely academic curiosity. It is usually a specific stuck point, and Algebra Tutor turns out to be the name for it.
The questions worth asking
Start where the understanding actually is rather than where the syllabus says it should be: If this question appeared on the exam in a different form, would you still recognise it?
Then test the reasoning rather than the recall: Which step in your working are you least able to justify? Two questions, asked in that order, do more here than a page of explanation.
Where people go wrong
Where this usually goes wrong is that re-reading notes is studying — a mistake that is invisible until something unfamiliar turns up.
Putting it into practice
In practice this means closing the book and reconstructing the argument from memory, then checking only what failed.
You will know it is working when the student can predict which part of a question is the trap.
Where to go next
If this is the thing you are stuck on, start with the advanced guide on question ladders, then a session on the paper you are preparing for.