I remember having my attention drawn to the difference between spotting pattern and spotting structure and I found that quite useful.
In the classic type of task where pupils are asked to find the number of matchsticks, say, in the nth term of a growing pattern many will be able to spot a pattern in numbers in a table but fail to get a sense of the structure which is where the generality lies. That's why it is useful to ask learners to attend to the structure of a complex example (i.e. the 15th term) rather than build up the pattern one by one.
In Dave Hewitt's article in MT163 (the ATM journal) called "Approaching Arithmetic Algebraically" he makes what I found a very thought provoking comment concerning the algebraic statement 3x + 1:
"But let us not confuse the two words involved - algebraic and statement. The algebra is the work you do in order to get yourself in a position where could make a statement (such as the example given above where there is a way of seeing the matches so that the formula is just a way of expressing what is 'seen'). '3x + 1' is the statement part, and only hints at the algebra which may have taken place to make such a statement".
Do our learners do this kind of work when they are doing school algebra?