Which metacognitive skills/abilities are involved as students gain facility/knowledge in this domain? Think of an activity or lesson component that explicitly teaches one or more metacognitive and one or more problem solving skills.
If I were teaching a lesson about parallel lines (which I recently did) I would expect students to continually use metacognitive skills. Students have to learn to continually check themselves and express their thought processes and answers to questions.
To begin, I would make sure students had an understanding of parallel lines by applying it to a real life topic, such as parallel bars in gymnastics. After we had covered and learned the definition for parallel, I would have students explain why parallel bars in gymnastics are called parallel. This would allow students to verbalize their thoughts and also problem solve.
I would also have students participating in overt learning strategies, that is, drawing parallel lines on a paper. These are thought process that we can actually see. I would then have students list what made the lines that they drew parallel, another observable behavior. This would also aid students in increasing self-explanation.
To increase problem solving skills of heuristics, with no step-by-step instruction, I would show students two lines that are not parallel and have them explain how they would manipulate them to make them parallel. This would cause students to really have a grasp on the concept and be able to problem solve.