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Solving (multistep) equations:
Unconsciously incompetent: Students just guess numbers that “fit” into the equation. Often students don’t know how the value they found is right or wrong. Guess and Check
Consciously incompetent: Students know they should follow a certain process but don’t remember it so they default to guess and check. However, I feel these students can tell about their thinking.
Consciously competent: Students know that they should be using opposite operations on both sides of the equals sign. They can “unpack” the equation to get the variable alone by “undoing” the parts given in the equation by working backwards.
Unconsciously competent: Students have internalized and follow the steps for “undoing” each operation in the equation.
Consciously masterful: Students consistently solve equations correctly (and know that they do) because they understand that only one value would make the equation true while all others would make it false. Given this they are able to check their answers. Additionally, they are able to use, or rather, not use opposite operations when not necessary (as in one step equations) because they understand they are looking for the input that will give you the desired output.