Study of ancient number systems can give students new arithmetic tools and a deeper understanding of numbers. Study of historical events that led to the development of the common mathematics that we use could reveal to students the usefulness of mathematics. They would learn that math evolved from a need to solve a problem. If radicals are involved in the solution of a problem, students should be encouraged to use approximations because an approximation is more useful in a real life application. It is more important to understand the concept of square roots and what it means to find the square root of a number.
Factoring is a skill that technology has made almost obsolete.
I am still undecided about how much factoring of polynomials should remain in the curriculum. With current technology, students can estimate roots of polynomials by graphing. Factoring in the past was the best tool students had for finding solutions. However, with graphing calculators, computer graphers, and the like, students have a much more powerful, and better yet, visual tool. Paper and pencil graphing is another task that technology has replaced.