Sure you can use the verb without "started", but you'd have to adjust it and you'll lose some information.
Recently at* our school, they teach computer philosophy.
This sentence doesn't make sense because "recently" hangs there without anything to give it context. So you'd have to change it to:
At* our school, they teach computer philosophy.
Now the sentence makes sense, but now you've lost the information that the teaching of this course has begun only recently. Even if you left out "recently", the word would be implied, as in:
They started teaching computer philosophy at* our school.
It's information about when something started (you could also say "they began to teach..."). So it's not just a matter of style or flow, it gives the sentence temporal context.
*Edit: Unrelated to your question, but I'd say that "at our school" should be used instead of "in our school" for this example.