My name is Tom Wirt, with Clay Coyote Pottery. I'll try to shed some light on the intricacies of clay cooking pots, especially tagines.
You can use any flameware tagine directly on the glass stovetop. This includes, Emile Henry, Le Crueset, and Clay Coyote flameware. These are pots with either a metal base (Le Crueset) , or a type of ceramic called flameware (Emile Henry, Clay Coyote which is formulated and made to take direct heat.
Normal stoneware clay pots and earthenware pots will not do this. Stoneware should never go on a direct heating source, gas, electric or glasstop. It will crack with or without a diffuser.
Earthenware ceramic pots, typically identified by a reddish clay color and some absorbency by the bare clay (typically the bottom), do need a diffuser and should be started over a low heat. They can crack if used over sudden or too high a heat. Remember that these pots were originally used over charcoal fires.
Metal, obviously is fine.
The flameware ceramic pots, are designed for direct heat and are actually especially good on glasstops as the top spreads the heat better than electric or gas.
Clay is a insulator, not a conductor. Thus the heat doesn't spread much, but, with a highly liquid food like a tagine, the liquid spreads the heat. Basically a tagine is cooked at a simmer, even though the pot would take the heat.
Induction stoves require a metal plate with ceramic cookware to turn the electromagnetic force into heat.
You can find more info on my blog.
Best Answer
Most of the dishes we cook do have at least some oils and fats in them. Even when we are not explicitly frying, or even sauteeing, the delicious fragrant vapors which fill our kitchens when we cook waft around and get into every nook and cranny of our walls, cupboards and stoves, even places where we can't reach to clean. These vapors contain tiny droplets of oil from our cooking. Over time, this deposits lots of grease in inaccessible places.
After a few years, even the blinds above the sink on the other side of my kitchen are lightly coated with a gross mixture of dust and grease. I do fry and sautee a lot, so it may take longer in your kitchen, but that's where that greasy residue comes from - through the air.