<p>Haven’t read all the posts but here is my 2cents:</p>
<ol>
<li><p>Math PhD are supported financially in full because all such graduate departments use TAs.</p></li>
<li><p>There are national and international fellowships that are prestigeous for the star students. Having one of those means that you don’t have to be a TA, which is a real chore. </p></li>
<li><p>There is a world of difference between pure math versus applied/practical math. While one may not know which research area to pursue, most applicants know where they fall. </p></li>
<li><p>I would not visit before invitation from schools. More useful will be to use the money to send him to a real math conference. He can talk to researchers there and hopefully see what it is really like to be doing graduate level math research in specific areas. </p></li>
</ol>
<p>Best wishes.</p>