RVSM Procedures
Your training captain is not just expressing his opinion. It is highly likely that your Ops Manual has some wording like:
"To prevent unwanted TCAS/ACAS warnings or alerts when first approaching cleared flight level and/or when changing flight level in RVSM airspace, ensure that, when nearing such cleared flight level, vertical speed is not excessive. With about 1500 feet to go to a cleared flight level vertical speed should be reduced to a maximum of 1500 fpm and ideally between 500 and 1000 fpm. Ensure that the aeroplane does not undershoot or overshoot the cleared level, manually overriding if necessary."
This is from our Ops Manual, but if I recall correctly, it is almost verbatim copied from the CAA's AIC on RVSM Ops. Because RVSM is a specific approval, and you need an approved RVSM Ops Manual, what is says in your approved Ops manual is "regulatory" and is not just "a good idea".