As... implies a comparison with something, and there isn't one, so you need 'by'.
By... denotes that something occurs as a result of something else.
Try swapping the clauses around, that usually shows up errors or the right way to put it.
Example of rearranging the clauses:
'As filtering the child's behaviour, the teacher's attribution is obtained...' doesn't work because one is left asking 'as... what?' What is going on at the same time... answer: nothing.
'By filtering the child's behaviour, the teacher's attribution is obtained...' makes sense because we see how the second clause depends on the first. How does the teacher's attribution get obtained? It gets obtained by filtering the child's behaviour.
'As' could work if you put 'as a result of' - which then has the same meaning as 'by', for example:*
'The teacher's attribution for a child's behaviour is obtained as a result of filtering the child's behaviour.'
By: 'Identifies the agent that is responsible for doing something':
- X is done by y
- Omelettes are made by chefs
- By adding up numbers we can do sums
- I get results by doing something
- By doing something, I get results
'By' is like firing an arrow into a target.
Wheareas 'as' is better thought of as running along, alongside a galloping horse - two things are going on or going along together, or being contrasted, or branching off in two directions, often being compared, thus:
- milk as well as eggs go into an omelette
- his omelette was as hard as a rhino hide
- he got the same results as her
- she looked up as he entered the room
If you leave out "the teacher's" then it reads:
'Attribution for a child's behaviour is obtained BY filtering the child's behaviour'
By is the fulcrum on which the two clauses are levered.
A is dependent on B - and B on A.