Other things to consider:
Parking is free with an AP (only matters if you are staying offsite, when it's $18 a day for parking, onsite get free parking).
OP....will this be your only trip to Disney in the next 12 months? If not, then consider that three 10days for a second trip would be another $1322....so then the AP would save you $600+ in tickets for both trips. A lot of people go "once a year" but plan their dates so that an AP will work...for instance you go this year from June 1 to June 14.....but in 2016 you go from May 15 to June 1. Your AP would cover both those trips (the AP is actually valid until the same date the following year).
A 10day hopper for 3 is $1322, a 4 day hopper for 3 is $710 additional, so you're at the AP rate as it is.
Also take into account that an AP will quite often get you a nice room discount....which might make up the difference. For April/May/June stays there is a 20% to 37% discount (depends on which resort, but basically values are 20-25% and deluxe are 30-37%)
Then throw in that AP gets discounts on merchandise and food (not everywhere, go read about the AP to be able to see if those discounts will work for your family). Those can add up quickly too.
OP will need to do some number crunching to decide if it will work for their family, but basically the "experts" agree that if your stay is over 10-12 days, then the chances are that the AP will be a better deal overall. Of course, the flip side of that is having an AP makes you more likely to decide to visit more often because "hey, we have APs!"....but of course that's a slippery slope.
What others have said is also very realistic.....14 days of nonstop park hopping is quite a bit and there is a lot to explore and enjoy without entering a park. And if you want to leave the site, Orlando has a great deal more than amusement parks too (and there is more amusement parks nearby even).
Whatever works for your family OP, I hope that you have a magical trip.