What is wrong with teachers teaching the concepts first and tutors reinforcing them, instead of the other way around?
My answer to that is that
sometimes the teachers don't do it well, or just do it wrongly. But both parties are culprits here. I've seen, and corrected, many tutors teaching the wrong thing.
I'm not sure where your students have gone tutoring in the past, but I've experienced the opposite! Many students tell me that they pretty much rely on my teaching to get them through their exams (and many perform quite well too) because their teacher did not explain things clearly. Now typically I'd be giving the teacher the benefit of the doubt because the opinions of students are generally to be taken with a grain of salt, but there have been (too many) times where I see the notes that the students have copied down from the board at school, and there are things that are downright wrong.
They can range from relatively minor things like
- "If the polynomial has real coefficients, then it has non-real roots in conjugate pairs and visa versa"
to more serious things (this one was recent) like
- Using normal 3U induction to solve strong induction problems.
- I had a recent example where there was a half-yearly exam question that couldn't be done without advanced university-learned techniques. I sent a counter-example to the student, who forwarded it to their teacher, and it was rejected (I was flabbergasted, how can you 'reject' a clear counter example?) I gave another and they finally removed the question from the paper.
- Last year when teaching Mechanics, many of my students had drawn centripetal force on their diagram. I asked them "Why?" and the typical response was "Well we draw the normal force, the gravitational force, so we also draw centripetal force".
- A few weeks ago, I was teaching recurrence relations in Integration and when dealing with the more difficult ones (which have quite a different approach to the standard reduction formula problems, so things like
), some of my students asked if these will ever be in exams because their teacher didn't show them any such thing. I directed them to 2013 HSC.
- Even more shockingly, when going through the problems where IBP was not required to construct the recurrence, some students were confused and when I investigated, I learned that they had been taught to ONLY use integration by parts! I directed them to 2014 HSC.
One of the things that I feel quite strongly against is 'super accelerated tutoring', where Year 9-10 students learn Year 11-12 concepts. Like you said, all they seem to get out of it is "Oh I remember seeing that formula".
In an ideal system, we won't need tutoring. But the key word here is "ideal".