The illustration at the top where sticks are used immediately made me want to provide the "outside the box" solution of arranging the sticks in 3D. But I guess an octagon is pretty clearly defined as a planar object.
Something the article didn't touch on but that I think is fundamental when trying to construct proofs: Often, you can get further in a proof by trying to prove the opposite. Assume that what you're trying to prove is impossible and try to construct a counterexample to the original proof. You will reach a point where the construction can't go further - examine closely what invariant prevents you from completing the counterexample and you will happen across something that will advance your actual proof. Note that this need not result in a proof by contradiction - it's usually more about illuminating the way forward.
In a sense, the octagon problem is teaching exactly this step. That is, the "real" problem here (and the proof that the students should arrive at) is "prove that you cannot construct a convex octagon with 4 right angles". How do you do that? You assume that it's wrong and try to find a counterexample to the original proposition - a convex octagon with 4 right angles. The ensuing "productive struggle" leads exactly to the insights you need to prove the original proposition, as detailed in the article.
Something the article didn't touch on but that I think is fundamental when trying to construct proofs: Often, you can get further in a proof by trying to prove the opposite. Assume that what you're trying to prove is impossible and try to construct a counterexample to the original proof. You will reach a point where the construction can't go further - examine closely what invariant prevents you from completing the counterexample and you will happen across something that will advance your actual proof. Note that this need not result in a proof by contradiction - it's usually more about illuminating the way forward.
In a sense, the octagon problem is teaching exactly this step. That is, the "real" problem here (and the proof that the students should arrive at) is "prove that you cannot construct a convex octagon with 4 right angles". How do you do that? You assume that it's wrong and try to find a counterexample to the original proposition - a convex octagon with 4 right angles. The ensuing "productive struggle" leads exactly to the insights you need to prove the original proposition, as detailed in the article.