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? Because time and time again, students who come in to my class “knowing” this formula have no idea where it comes from. And because for the ways in which it is used in the first two years of high school, the Pythagorean Theorem is much more intuitive and direct. Kids never have a problem remembering from middle school that a2 + b2 = c2 for right triangles, and if you ask them to find the distance between two points by drawing an appropriate right triangle and making calculations, they can do it easily.
But ? That equation is the epitome of a meaningless formula for most kids, something they have to memorize because they are told they have to “know it”; unless it is emphasized for them, they don’t even remember that it is derived from the Pythagorean Theorem! It introduces abstraction and symbolic manipulation when it isn’t needed or helpful, and it doesn’t let them solve any problems they couldn’t have solved otherwise. It isn’t until 11th grade or beyond for most students that solving for d symbolically seems appropriate to me in terms of their level of understanding, where they can appreciate it (such as when they solve loci problems or explore conic sections).
Let me put it this way. Something seems wrong to me when students think in an algebra course that finding a perpendicular slope is easy (“I just take the negative reciprocal and I’m done”) and finding the distance between two points is hard (“What is that weird formula again? Why do I subtract the x’s and y’s, square them, and then add them together? I have no idea, but I know it will give me the distance, so here goes” ), when in fact in terms of their actual understanding it is usually the reverse. Even though they find it easy to compute a perpendicular slope, students often have no idea why it is the negative reciprocal; in contrast, despite having no feel for the distance formula, most, given two points on graph paper, could explain how to use the Pythagorean Theorem to find the distance between them.
Don’t get me wrong, I’m all for introducing symbols and abstraction when it makes sense to do so and is at least partially motivated. For example, when one is proving the law of cosines, solving for d is clearly the right way to go, and students will quickly see that. But I think many kids decide they don’t like math at the point when it starts to seem arbitrary and obtuse instead of a way of answering questions they naturally have. I don’t see the Distance Formula answering any of their questions, and so I wonder– why teach it to them?