Always_Ask_Questions,
” the methods tried during that period was to start at teen ages (and the result was that they did not listen to parents). So, they were not following Pirkei Avot.”
I think the reason they started teenagers in the second takana listed was not because they were experimenting with the best age to start learning Torah – they knew it should start at a younger age – but rather because the schools were set up “pelech upelech”, one in each larger district (comprised of many smaller districts/towns) as opposed to one in each town. Therefore it wouldn’t have been safe for a young child to travel a long distance to get to his district school.
Another possible way to look at it – perhaps the takanos weren’t trial and error, but rather progressive responses to a deteriorating situation. The first takana bringing students to Yerushalayim – perhaps they were learning Torah with their fathers but increasing numbers needed the kedushah of the city to get to the higher levels of learning that previous generations attained. But then growing numbers of fathers weren’t even able to bring their sons to the level needed to attend in Yerushalayim, so they set up the district schools for teenagers to learn “gemara” after presumably learning the alef beis, chumash, and some mishnayos at home But the boys did not have the proper discipline to accept rebuke from a new teacher, so finally Yehoshua ben Gamla decreed to put teachers in each town so young children could attend safely and be exposed to teachers outside of home?