Why use algebra when you can use arithmetic?

This post was jointly written with my colleague, Steven Francis. It follows from my earlier post, Welcome to Maths class: Leave your common sense at the door. In this post, we provide examples that highlight the importance of allowing students to solve problems from first principle. At times, insisting on specific formal methods can diminish students' ability to solve problems.

In the video above, the renowned physicist Richard Feynman relates how he could solve linear equations early but was told "You did it with arithmetic. You have to do it with algebra."

Feynman then reflects with visible annoyance:

There's no such a thing as you don't do it by arithmetic, you do it by algebra. It's a false thing that they had invented in school so that the children who have to study algebra can all pass it. They had invented a set of rules which you follow them (sic) without thinking to produce the answer.