Another issue is that Math is a very snotty field and grad school recruitment is specific. Texas tech is a very good university and its honors college is excellent but may be what a professor called “a handicap to overcome” for PhD admission.
Look at how many math majors graduate per year, in relation to the number of students enrolled, and ask how many enrolled in a PHD program. You can also find the origin of actual awarded PhD.
On another front, Vanderbilt costs 130k over 4 years, of which 80k are covered. 50k= if he wants Vanderbilt, he’d be to work part time from now till graduation and full time all summer (=4k) as well as 8-10 hours a week (2.5k per semester) + take his subsidized loans (5.5k), and achieve this level of summer/academic year earnings each year. Is that doable? Or are the 50k AFTER a financial aid package that includes a federal loan and work study?