Richg is absolutely correct. But all ice damning is not necessarily due to heat loss &/or air infiltration in attic. I agree with him, check for air leaks, gaps etc. Fix these if you find them, but most ice damning occurs merely from the days sun melting the snow/ice on top of the roof surface running down and refreezing as the day goes on and starts getting colder again toward evening. This is where the ice & water shield comes in play. So yes eliminate heat loss problems in the ceiling under the attic, The attic itself is made to vent, ridge vent, vented soffit, gable vents etc. You want air flow in the attic cavity, but not from the heated interior of the house.
Again, as Richg said, eliminate heat loss from living space below attic, but you may also need to get ice & water under the eave edges to combat normal roof surface melt/freeze caused ice damns. Not to down play heat loss into an attic, as it is a major energy waster. But in a vented attic, the nominal heat loss into a cold vented attic is not the majority for causing ice dams on the roof surface. It would take almost a furnace blowing directly into the attic to heat the air & underside of th decking enough to cause this type of problem. And in a vented attic, usually what nominal amount of warm air that does leak into the attic is diluted and cooled much too fast to warm anything up there or make much of a temp change. Now if the attic is not vented properly & is tight, dead air, then you could have a problem with temperatures raising some in the attic space. No attic/roof system should be air tight, if it is, you will find the underside of the decking black, moldy or even rotting from the underside duw to moisture & heat build up in warmer/hotter months and no air circulating in the space. I have seen this countless times, especially with cathedral ceilings, where the insulation was laid to fill all the way up to the bottom of the plywood or whichever decking they used. No air flow= rot, mold, and premature aging of the decking. And also prematurely shortens the shingle lifespan.
Most damning occurs down near the gutter edge, valleys & curbs. Most residential structures do not have curbs, but most have gutter edges & quite a few have valleys.
At the gutter edge the days runoff refreezes in the gutter, building up ice to above the top and as the runoff comes down it now starts hitting this damn and running running back behind the gutter and then sooner or later up under the shingles. Another reason for vented soffit, especially extended soffit, is if the damning does occur at the gutter and runs back in, it at least runs down onto the vented soffit and drains back out before getting back into the wall. Unless the soffit is pitched towards the wall. Not saying you would want this to happen, but better this then finding the wall and running into the wall.
Same can happen in the field of the roof, run off freezes, creates a damn, next run off comes down, hits damn refreezes back until it gets under the shingles, until it backs up enough to hit a seam underneath, and sooner or later the runoff finds the seam and leakage occurs. Usually in the field this is due to a low spot in the decking, a slight bow is sometimes all it takes. But the majority of ice dams & leaking caused by them are at the eave (gutter) edges, and withing the first couple feet. This is why ice & water shield is spec'd on almost every home built these days. Ice & water is a roll of sticky asphalt self sticking underlayment made specifically for this purpose, hence the name. When the shingles are nailed down, the nails do penetrate the ice & water shield, but the asphalt of the material seals around the nails. So if installed correctly, which a lot of places do not do, even though damning may happen, it typically does not back up over 3' and will harmlessly run back down when it melts. Ice and water shield should be run the full length of the eave, and the edge should come down and onto the fascia or face board an inch or so. So when the gutter is installed, the ice & water shield is below the rear gutter top edge an inch or so and sandwiched between the gutter & face board. Now if the gutter damns up and water runs back it has no where to go but down between the gutter and ice & water shield once it melts. If you just run it to the edge of the decking, it can still run in at the seam of the decking & fascia board. Keep in mind alot of ice & water shield products are not made to withstand direct sunlight for any long period of time. So just sticking it on and not covering with something will enable it to deteriorate in time. There may be newer products that are resistant to this, but in most cases, the ice & water shield should not be showing or exposed in the first place if done correctly.
Bottom line, seal air leaks & install ice & water shield, and be done with it. Or in time things will rot and even more work will have to be done.