One of the things that can be hardest for kids learning algebra is to be able to understand the value of abstraction and of using symbols to help one analyze and think about a problem. I myself remember learning algebra (way back in 1978) from Dolciani/Wooton, a textbook that valued formal manipulation above all else; there was almost no motivation given for where the rules came from, just lots of practice in learning how to manipulate symbols correctly. Indeed, for a number of years afterward I thought that formal manipulation was all there was to algebra.
The idea that there are ideas to be discovered in algebra was completely foreign to me. I knew in 9th grade that perpendicular lines had slopes that were negative reciprocals, but if you had asked me why, I would never have known, or even thought I should have known. Dolciani gives a proof, but I would bet a lot of money that very few students read it, as it is, to be frank, inappropriately abstract for a high schooler learning the subject for the first time.
Our curriculum ( http://parkmath.org/curriculum/ ) approaches the topic by having students first draw a line on graph paper with a slope of 2, and then try to figure out experimentally what slope a line perpendicular to it should have. Working out concrete examples using lines of different slopes can lead to a much deeper understanding for a 9th grader than an algebraic proof that they have little chance of following, never mind retaining. When students are older and have more experience with algebra, of course, a formal proof can make good pedagogical sense. But for a freshman?
Which is why I think we should ban(!) teaching the “Distance Formula”, at least for the large majority of 9th and 10th graders. Why, you might ask, if I am trying to teach them the value of abstraction? Read More