This article summarizes the current debate — call it a “controversy” — about brain training. (The authors prefer the phrase “cognitive training.”)
The authors conclude that intelligence can be increased, but … so far … only in controlled lab settings. That is: NOT in schools with various training programs…
This article does get a bit technical; for instance, the authors debate whether or not a particular series of studies ought to have been included in Melby-Lervag’s & Hulme’s well-known meta-analysis. (Melby-Lervag & Hulmedid not include the studies, and these authors think they should have.)
At the same time, I think the complexity of the question is the point. If you or your school plan to devote the considerable time and money that this kind of training requires, you should know all sides of the argument.
(Caveat emptor: this article is, in effect, an advertisement for the authors’ book. I haven’t read that book, but … based on the thoughtful balance of the article … they strike me as sensible folk.)