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 9^{th} 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 *a*^{2} + *b*^{2} = *c*^{2} 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?

## 18 Comments

The distance formula does seem pretty useless to me, unless you are trying to prove a given locus formula, or something else reasonably advanced. The Pythagorean theorem is much easier for students to use. I think that the distance formula could be introduced once the students are capable of deriving it themselves, which may vary from group to group.

Great Post Tony!!! The reason why we don’t “Ban” it is because it’s easy to have kids plug in numbers and solve problems with it. Besides isn’t that is what we ask them to use on the SAT, IB, AP tests don’t we? As you imply too young and way before we have taught them the conceptual understanding of it first. So they go through math with all these formulas and tricks with little to no conceptual understanding of the concepts or big idea. When we are not clear about the conceptual understanding of what we are teaching we tend to teach only knowledge, skills, tricks and shortcuts. Where is the thinking and understanding?

I have a couple of things I would like to “ban” as well.

Mid point formula – Don’t we teach what average and mean is long before this?

Foil method – WHAT??? I thought this was distributive property (or rainbow method)

In high school, from ’65 to ’68, I am sure I had no clue that the Pythagorean Theorem and distance formula were connected. When I became interested in mathematics, in the early 1980s (it took me over a decade to get the bad taste of high school math out of my mouth, though luckily I’d had no college math to leave a similarly foul flavor), I made the connection myself. As a mathematics teacher and coach, I never teach the distance formula any more and advise those with whom I work to avoid teaching it. Eventually, as you’ve suggested, Tony, students can derive it themselves and it has meaning.

But it’s not actually true that the SAT or ACT require knowing the distance formula. In fact, for most situations that I’ve seen on those tests (and I have seen so many such exams that I long ago lost count), it’s more important to be able to quickly apply the basic ideas of the Pythagorean Theorem and the use of triples to find needed distances. Quick mental math and recall of a few triples and their reasonably small multiples (not to mention knowing, for example, that there is no triple that begins with 1 or 2, so that any right triangle (or distance) involving a leg of either number MUST have a radical as the result, assuming the other leg is also an integer) will pay big dividends on these exams: memorizing the distance formula? Not so much.

As to Dan’s comments above, I have to restrain myself from physical violence when one of the teachers I coach blithely reminds students of FOIL. I have preached against that particular bit of mnemonic nonsense for several decades, and it absolutely enrages me that even my Dolciani-using 9th grade math teacher saw nothing wrong with teaching it to us. Though certainly someone who should have known better, she did us the disservice of bogging us down with something that, personally, I found ridiculous and never made use of (though not without the feeling – one I often got in school mathematics – that I was somehow ‘cheating’ by just reasoning my way through the process, i.e., using the distributive property). And of course, all it takes to explode the FOIL bubble is to ask the instructor or student to multiply a trinomial by a binomial. No mention of FOIL need be made for students who are FOIL-dependent to become hopelessly lost. If one is lucky, though, a small sign of enlightenment may be emitted from the odd instructor capable of reflection.

And may I add to the list of student question,

Why is x2 first and x1 second?I agree that the ‘distance formula’ is being taught in the wrong place. When students are asked to find “the” distance between two points using the formula, they are participating in a broader notion of “metric” and the axioms which establish that area of study… but this is not their current area of study! Many of them will never encounter the concepts of metric spaces where it will be convenient to know the L2-norm formula so that a more general Lp-norm can be easily created; what the students are being taught (and what they need to know) is that the L2-norm is the ‘correct’ measure of distance for them, and for that drawing triangles and ignoring the naming of the formula I believe is enough.

On a personal note, I remember the day I was taught the distance formula: we were not shown its derivation, but the teacher was going slow enough for me to figure out I could draw the triangle and Pythagorean-it before she finished giving us notes. But, when she did give us the formula, it was somehow different from the one I created (probably subtracted x2 from x1, while I did it the other way), and I didn’t recognize until much later that they were equivalent, so I resigned myself to memorizing another math fact and stopped exploring on my own.

I don’t completely follow the post. Won’t simply guiding students constructively to develop the distance formula offer intuitive and algorithmic knowledge? The lesson is that the distance formula should not be taught in the traditional way, not that it shouldn’t be taught at all. Teaching student to derive it makes more sense. Don’t give up on a valuable formula and learning opportunity!

Aye.

I really like the point you make that methods and formulas developed and taught in class should answer students’ questions. I also think that a post-geometry algebra class should involve using algebra to answer geometrically-motivated questions and viewing geometric objects and situations through an algebraic lens (and vice versa). How about this for a question to ask in an algebra class that might motivate algebraic formalism of distance without introducing it for the sake of formalism: “If I give you a point, can you develop a method for finding the points a particular distance from that starting point?” Kids don’t need the distance formula to find the distance between points, but they’ll probably want and need something algebraic to find points a particular distance from another point (if those points aren’t on the horizontal or vertical, or if you want to describe all of them). Such a question (and other types of locus questions) would certainly have to come a good deal after students are already comfortable finding distances between specific points using the Pythagorean Theorem, and probably after they’re comfortable answering the question in one dimension.

I am late to this post…

I do try to lead my kids to notice that the distance formula = the pythagorean theorem = the formula for a circle. I also point out the relationships between distance on a number line, distance int he cartesian plane and distance in 3 space. In and of itself, it is a useless formula. When you help uncover connections then some math can stick.

I also totally endorse the banning of FOIL. I simply refer to it as that four letter F word that I will not use.

I totally agree! So many students…and sadly, teachers…don’t see the connection between the distance formula and the Pythagorean Theorem at all. That is a shame. Learn one formula (the Pythagorean Theorem)…in a constructivist manner…and then apply it to multiple situations…distance, equation of a circle, law of cosines, trig identities, etc.

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like to send you an e-mail. I’ve got some creative ideas for your blog you might be interested in hearing. Either way, great blog and I look forward to seeing it grow over time.

I always taught that it’s nothing but the Phythagorean theorem, and that if they grasp that concept, there’s nothing to memorize. I consistently used conceptual “tricks” that once understood, made remembering a “formula” easy if not impossible to forget. Some students still prefer to memorize, but many understand. I don’t necessarily think that eliminating the distance formula entirely is the best approach since they’ll be expected to be able to use it in schools using other curricula and at university, but it can certainly be used to encourage students to understand the underlying concepts versus storing something to short-term memory for a test. There’s really nothing to memorize if they just get the concept.

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