Why do students struggle with geometric proofs?

We've spent a couple of days working through puzzle type problems that involve angle relationships, both triangular and angles formed by lines (intersecting, parallel, etc.). Students have been engaged in working through these though they still struggle to different degrees. There is a tendency to not see what is given, to not think about what is given actually means, and to not use what has been solved as a piece of the next solution. After working through challenging problems for a couple of days they are doing better with these.

The other piece that continues to challenge students are geometric proofs. Since we've been working on problems involving the exterior angle theorem of a triangle, I thought this would be a simple proof for students to tackle.

I presented the scenario, with diagram, and asked them to describe how they could prove the following theorem: The measure of an exterior angle to a triangle equals the sum of the measures of the two non-adjacent interior angles of the triangle.

Walking around and talking with different groups and then discussing as a class showed they were thinking about the proof in a proper way. Students described that the exterior angle and the adjacent interior angle formed a linear pair and were supplementary. They explained that the three interior angles summed to 180^o. They said you could substitute and subtract to show the result. They could say all of this but they couldn't write it out.

As I walked around, I could see that students would leave steps out. How do you know that those angle measurements sum to 180^o? Why do you say these two angle measurements equal the exterior angle's measurement?

Some of the struggle comes from a lack of understanding the vocabulary. You cannot write a proof if you don't know what a linear pair is and what properties it possesses. Part of it is that students do not have experience in writing step-by-step detailed instructions. Writing a proof is akin to writing a computer program. You cannot skip steps or the computer won't understand what you want done. Proofs are the same. You can't skip a step or the reader is left scratching their head as to how you got from step A to step B.

We're told that students should have a level of proficiency in writing proofs by this time in the semester. I don't think that is realistic. Without having the background of writing detailed step-by-step instructions, students do not have any basis upon which to draw.

It is necessary to build this basis of writing detailed instructions that can lead to proof. I will be experimenting with some ideas. For example, have students construct a figure with compass and straight edge and then write instructions so that another student can replicate the figure. We are moving into congruence next. so I'll be able to test this out to see if it helps.