I don't know what to believe regarding the mashing vs. spinning theories. There seem to be two schools of thought that I have read about.
Now before anyone flames me for what I am about to write, I just want to say that this is what I have read (others wrote it), and I have also read some who are for spinning.
And, again I don't know who to believe. I have tried both, and my average speed seems not to differ. However, mashing makes it easier for me to stay in a higher gear longer, and this helps propel me up hills better. Spinning makes me keep downshifting to a level of comfy spinning up the hills, and this slows me down on the ascents. So, for now, I am back to mashing, not circling through the stroke.
I have read that, in studies of pro riders with electrodes or something attached to their leg muscles, they are not really powering thru 360 degrees of the pedal stroke, but just mashing on the downstroke and lifting the foot that is going into the upstroke so that it doesn't fight against the foot going into the downstroke. But there is no power through the 360 degrees of pedal circle. They say none of them are doing that. Just mashing and lifting.
I don't know if this is true, or if what the pro spinning peeps say is true that you can power thru the whole pedal stroke.
Try both approaches and see what works better for you.