> i. Would it be sufficient if they proved they understood what addition, subtraction, multiplication and division was? Or do they need to be able to correctly calculate what 4592 * 314 is? What if they could show you using diagrams of squares and rectangles why (a + b)* 2 expands to the equation it does but refused to chart out results for various values of a and b?

I’m not an educator by profession but I’ve done a lot of training and mentoring.

There is no replacement for actually doing the work. People who study something but don’t go through the exercises feel they understand topics better than they do. It’s only when they are put in a position where they have to apply it that the cracks in their understanding are revealed.

This would 100% result in students who thought they could explain multiplication but only had a surface level idea. The knowledge would also be fleeting because actually doing the thing makes the knowledge stick more than just observing and understanding the thing.

So yes, you still need to have the students do the thing.

Replace math with writing and it will be more obvious. It’s easy to look at someone’s writing and critique it. It’s much harder to write well without practicing.