This video is called “Japanese multiplication” by the user who uploaded it to YouTube, and that identification has been used by Gizmodo, Brainpicker, and others. But its source is unclear; the technique may be drawn from a body of mathematical concepts called “Vedic mathematics.” In fact the same video has been uploaded to YouTube as “Vedic multiplication” as well, and videos in the same style can be found at a blog called Vedic Maths Forum. Vedic mathematics is a system promulgated by a 20th-century Indian spiritual figure named Bharati Krishna Tirthaji, who controversially claimed Hindu scripture as the source of a canon of interesting mathematical tricks and algorithms.
The technique is demonstrated—using the same numbers—in a more recent video by by author Alex Bellos produced in conjunction with his book Alex’s Adventures in Numberland (April 2010):
As video, these aren’t tours de force like Vi Hart’s doodling of infinite series (someone give her a book contract, please), but the conceptual scheme underlying it is pretty mind-blowing. While I don’t think it’s any faster than the kind of place-based numerical calculation of traditional methods, the mathematical imagination it suggests is radically different. It’s not clear where it originates, though; maybe someone can tell us? In the meantime, to Gizmodo, Brainpicker, and the rest of us bloggers, the message is clear: like your math teacher told you, it pays to check your work.