Arduino Uno and Dual Step Motor Shield 1.1

My Arduino board allows the step motor shield to piggy back, first question is do I have to power both, I am getting my 12 V from 6AA batteries driving my stepper motor. From all of the examples I have seen I have noticed that the motor is being driven by the Arduino board my second question is what is the purpose of the motor shield. I have tried powering both boards through the motor shield using 12 V dc adapter 3.0 amp, obviously a bad choice as the motor shield got really hot but still working. Which brings me to my last question will 12 v coming from AA batteries have enough power to run this motor 2.8v 1.68a 4.4kg-cm RepRap motor. I am trying learn as I go my first objective is to get this little setup to output 30 rpm and 1 rpm I will have to figure out how with a press of a button I can pause the motor, with another button to activate a rewind at 30 rpm and forward or run at 1 rpm, any help I will be very happy with.

All motors need a driver of some sort to provide the necessary current and power - Arduino pins have an absolute
max of 40mA, not enough for a motor (unless its a really really tiny pager motor).

That stepper would work best from a chopper-driver, not a straight H-bridge motor driver. The classic one is the
Pololu A4988 board but there are others like the EasyStepper. Basically any chopper-drive board with microstepping
will give you decent performance and take less current from the supply.