I think it would help to achieve a synthesis of mathematics and history. I do not mean how the development of mathematics influenced history(or, on the other hand how various historical phenomenon influenced mathematics), but how mathematicans acted as historical figures. It would be interesting to know how mathematicans communicated with each other, how cooperation helps to engender beautiful mathematics and how inevitable disagreements follow from different personality and world-view. Controversies in mathematics, though rare, helps to clarify issues that made further development possible.
It is not possible to make all students get used to mathematical thinking, for many of them must come from a varied background having little to do with science. Instead one should focus on the human side of the story to let them believe mathematics is a subject that focused on original(not magic or mechnical) contributions, sharp individual insights, and the mathematican community is no different from other professional groups of natural science. Instead of showing 'how spectacular math is!' and drill the students with introductory level texts that marvel them, one should help them realize how mathematical research is being produced in real life. For example, it is often being misunderstood that mathematicans do not make experiments; that sitting all day in front of a computer doing huge amount of calculations is the typical way of making progress. One should make the student understand the process of constructing a theory that might help us understand some underlying structure.
It is equally important to let the students understand that mathematical proofs is not so different from other forms of formal logic employed in real life. From my experience mathematicans often suffer from the shallow opinion by the public that they are being too critical on details and could not see the whole picture behind the main argument. It important to let them realize that while insisting on consistency and simplicity, mathematical proofs are ultimately human-mind products like the process of using rational reasoning to reach a conclusion in other fields.
One may organize a reading course by mimicking Carl Siegel's seminar at a much lower level. There are plenty of historical manuscripts, important papers, etc that has already been translated into English and worth discussing in a class. Manuscripts I can come up with are:
Abel's proof of the impossibility of solving the quintic equation in radical;
Riemann's speech "On the hypotheses which lie at the foundation of geometry";
Siegel's book "Topics in Complex Function Theory", Vol I should suffice for the purpose.
E.T.Bell's book "Men of Mathematics".
Hao Wang's book "A Logical Journey: From Gödel to Philosophy".