The last goal the projects directly address is
Here again, it is clear that they do mathematics, and that they see this. Both in the footnote by the Euclidean algorithm group quoted earlier about them having done mathematics, and in other statements by students. For example, Alan said:
It was all about how do you do mathematics.
In terms of what the relationship was between the class and the projects. Others also felt that the experience was beneficial. For more on this, look at the link in relationship of projects to class under the ten cards project. But perhaps this was best said by Ron:
But I am still glad that there was a project. I mean this is more general than the questions you want. You were asking. I am glad there is a project because in none of my math classes have a project. I don't have the opportunity to do that like in any other field they do. We have a project so I wish we had one other project somewhere else in math. Exercises different muscles or different practices I don't get to exercise.
In other words, Ron is pointing out how in mathematics you rarely have the chance to do the sorts of things the researchers in the discipline do, and moreover, that it requires different skills. Thus, I don't know that he would say that he learned a greater understanding of how mathematics is done, but I think it is safe to say that he felt he gained a greater understanding about how the mathematics that is done by mathematicians is done.
Finally, from all that has been said previously, it is clear that students gained more knowledge about the defining, refining, and exploring processes that take place in mathematics. For example, see the analysis of the Euclidean Algorithm Project.