How much importance do pure mathematics graduate schools give to programming ? Like how much programming knowledge do they expect a successful applicant to have ?
Not a math grad student, but I did take one graduate math course in discrete math. Programming was not required, but being proficient in Mathematica or Maple made everything much easier.
There is a range of expectations that more or less lines up with the expectations and needs of the faculty at the institution. If half the faculty require programming skills to complete a PhD under their guidance, then the lack of programming skills will be a barrier.
Also, there is an evolving understanding of what "programming" means. Some mean more traditional languages like Fortran and C. Others include skilled use of higher level tools like MMa, MATLAB, etc.
The fewer tools a student has in his box, the narrower their options.
I can't really recommend any scientist, engineer, or self-respecting math geek try and blaze an educational or career path without some programming skills in their tool box. Your options will be too narrow and only get narrower over time.
Yes, I agree that programming skills can only be beneficial for students in the sciences. Sometimes, it might even be mandatory for certain students. Many universities have a programming component as part of the requirements for the completion of the mathematics major. Let's say a student wants to go to graduate school for a certain sub-field in mathematics and let's say this sub-field does not use require much knowledge of programming. What if that student gets a B+ or A- in those required programming classes. Would a graduate school be willing to overlook that if he/she makes up for it with his grades in other math classes ?