This recipe comes from a very special lady (my mother) who was a nursery school teacher for more than 20 years. For years she cooked playdough for her class every couple weeks, and because it smells kind of yummy while cooking, it was always a disappointment to find out she was actually making something inedible. Still, making your own playdough does without the waste of manufacturing and packaging; plus, it's cheaper.
On the stove, use a large soup pot or dutch oven. Put heat on medium. Mix flour, salt, and cream of tartar in pot; slowly add water and stir like mad while cooking. Slowly add oil and keep stirring. As the mixture cooks, it becomes firmer and harder to stir. Keep cooking until it forms a cohesive mass. (Add food coloring with water early on, if necessary).
After the dough forms into an impossible-to-stir mass, dump onto the counter top covered in wax paper or wooden cutting board. At this point it will be hot, but pliable. Keep dough covered with a kitchen towel or wax paper to avoid drying out, until cool enough to store.
Store, when cooled, in an air-tight container in a cool place. (If stored in a hot place, the playdough will get sticky and unusable.)