Consider a lesson plan you might use. Which metacognitive skills/abilities are involved as students gain facility/knowledge in this domain?
One lesson plan that I have in mind could be a third grade science lesson about matter. If my students were doing an experiment determining how different conditions affect different types of matter, I would start the experiment with my students making their own individual hypotheses. After the experiment, I would have my students reflect on whether or not their hypotheses were correct. If they were correct, I'd have them explain why they came to that conclusion. If they were incorrect, I'd want them to identify why their hypothesis wasn't accurate. This would cause my students to think about their learning and help them understand why they came to the conclusion that they did.
Think of an activity or lesson component that explicitly teaches one or more metacognitive and one or more problem solving skills.
Any elementary math lesson can teach metacognitive and problem solving skills. Students are not only encouraged to understand and learn the math skill but also recall and explain how they came to their answer. Oftentimes students are given a math equation and asked to provide an answer and explanation. This causes students to think more about the step-by-step process of solving the problem as well as thinking about their thinking.