Kari -
I stand corrected, You are correct, the fan should work at reduced speed- tried a fan on a variable PS, Fan started spinning at approx 4 Volts. Got My DC motors crossed with AC motors. In AC motors if the RPMs drops to low the counter EMF is to low and the windings will overheat.
If the fan has LEDs they would be dim to not on.
Ref how to connect - any gorrilla can do it easy. - this is not the first, one poster wipped out his HDD.