You're describing exactly the situation in a lot of maths ed. Teach them conceptual stuff or teach them algorithms. The correct reply is, of course: both. There are even papers called things like "The Math Wars" going on about this. I used to be more on the conceptual side, but I'm getting all stern and schoolmarm-ey in my old age and am tending towards "learn your #&*@$% times table!" etc.