One thing that we might have done wrong is to have connected a 3V stepper to the Adafruit motor shield and using a 5V supply. But on further investigation this might not be as simple as that.
According to the data sheet, the 57BYGH420 is a unipolar stepper. However, a page Google found suggests that 6-wire steppers can be connected either as a unipolar or bipolar motor. (It also tells me that our motor is a NEMA 23.) Indeed, according to Stepper Motor Connection Options on piclist.com this is indeed the case, and to be precise 6-wires can be connected either as unipolars or a “bipolar-series.”
What’s interesting is that what Angela did to the two centre taps was in fact correct (whereas Adafruit’s documentation is not): To configure a 6-wire stepper as a bipolar, the two centre taps should “not [be] connected to anything.” That is probably why when I tried connecting the centre taps to GND (as per Adafruit’s instructions), the motor stalled.
It is also worth noting that the page says that if we connect a 6-wire as a bipolar, we get twice the resistance than as rated and 4 times the inductance, and require 70.7% the current and 141.4% the voltage. Which would mean the 57BYGH420 as a bipolar would be actually a 4.23V motor at 1.414A. So it in fact needs a supply that can deliver approximately 5.43V. Our 5V supply is low, but not by so much. A 6V, but not a 9V, will probably do better.