According to Vedic Maths India, Trachtenberg developed his system when he was a prisoner in a concentration camp in World War II. In 1950, Trachtenberg founded the Mathematical Institute in Zurich to teach the system.
The system provides an arithmetic shorthand through algorithms or, as Speed Math calls them, "a series of keys that must be memorized." The site claims that "arithmetic becomes delightfully easy because you will be able to 'read' your numbers" rather than having to solve the problems using traditional methods.
The system teaches speed addition, speed multiplication, speed division, squares and square roots, all of which a student of the program will "learn how to visualize" mentally and thereby "develop powerful concentration," according to Speed Math.
Speed Math's online biography of Trachtenberg explains that he "chose children who were doing poorly," and the students showed immediate results. He even instructed a 10-year-old mentally-handicapped child, who subsequently "acquired excellent memory habits" and an increased "ability to concentrate."
In 1960, Anne Cutler and Rudolph McShane published a translated version of Trachtenberg's book, which is widely available for purchase in both bookstores and websites such as Amazon. Speed Math also offers software based off of the Trachtenberg system that allows for additional practice.