We’ve all got black boxes in our lives.

A black box is a little mystery that you take for granted. It’s something you use without thinking, without skepticism, without once opening the lid to peek at the workings inside. For all you know, it might be powered by wind, water, cold fusion, hamster wheels—or even some fantastic combination thereof (i.e., nuclear hamster windstorms).

I’ve got a lot of black boxes. Too many. Cars, computers, non-stick skillets—every day is a decathlon of technologies and tasks I don’t actually understand. That’s why I love math, which (ideally) has no black boxes. In math, you only use tools once you’ve studied every gear and lever so closely that you could assemble them in your sleep. Math is the realm where all boxes are transparent.

Ideally.

This is, in fact, the exact opposite of how most students experience math. Too often, math is an inscrutable recipe book, describing foods you’ve never sampled (and probably wouldn’t care to). You must follow each recipe to the letter, because you have no clue how the ingredients taste, or what will happen if you combine them in new ways. And when you finish cooking, you throw the meal in the garbage disposal, and begin again.

For many students, math is just a big pile of black boxes. It’s like the worst Christmas ever.

Here’s an example. When I taught Geometry, the 9^{th} graders arrived cheerily able to recite “the distance formula”:

It’s for finding the distance between two points in the xy-plane. For example, say your points are as follows:

Then, if you’re one of my 9^{th}-graders (now college freshmen—my babies are so old!), you plug and chug:

The process is like kissing a robot: cold, mechanical, and not particularly romantic. You’re left wondering what’s going on inside. There is, of course, a better way to learn this. You begin by connecting the two points:

To find the distance between those points means finding the length of that line segment. But how do we find that length? Well, we furnish that line with some legs:

Look at that—a right triangle! And how long are those legs? Well, the vertical leg stretches from 2 to 6, so it has length 4.

And the horizontal leg stretches from 1 to 4, so it has length 3.

What’s next? Those who know the jungle will tell you that where you find a right triangle, the Pythagorean Theorem often lurks, too.

This is more satisfying if we’ve already proved the Pythagorean Theorem, but even if we must accept the Pythagorean Theorem as a black box of its own, that’s okay. It’s still worth breaking the distance formula’s engine into components, even if we don’t fully understand each part. Either way, solve the equation that Pythagoras furnishes, and look what we have:

This more natural method agrees with the formula, and doesn’t lead us through a dense thicket of mysterious notation.

Do enough of these problems, and you start to see patterns. Consider the distance from (1,1) to (2,5):

We don’t actually need to draw the triangle. We just need to know the length of the horizontal leg and the length of the vertical leg. Then we can apply the Pythagorean Theorem. We can even do it with generic points, using variables that could stand in for any coordinates—call them (x_{1}, y_{1}) and (x_{2}, y_{2}).

We consider the horizontal leg:

Then the vertical leg:

Then we apply the Pythagorean Theorem:

Then we solve for distance, and what do we find?

Yup—the original formula.

We’ve now built the distance formula box for ourselves. It’s transparent. Should we ever find we’ve “forgotten” the formula, we can quickly redevelop it, because we know exactly what’s going on inside. It’s powered by right triangles and Pythagoras’ theorem.

When it comes to distance, this might not change the way we compute, but it’ll change the way we think. This approach hints at how we might find distances in 3D, or distances along a curve. We’ve bought a better grasp on the formula with a single down payment. With any luck, future computations will flow more swiftly and error-free than if we’d adopted the formula arbitrarily, without rhyme or reason.

Best of all, we’re learning how boxes—all boxes—are built. We’re growing ready to build little boxes of our own, which is the loftiest goal of the mathematician.

As for the other black boxes in my life—cell phones, Bank of America, all-in-one shampoo/conditioner—I should probably investigate them, too. After I generalize the distance formula to *n* dimensions…

*Thanks for reading! (And for not asking why the pink alien’s shirtsleeves are so long.) You might also check out *Stupid Graphs!*, *Following Recipes*, and *The Quadratic Formula Must Die! (or, Long Live the Quadratic Formula!).

like kissing a robotOf course, there are robots, and then there are robots … and then there’s Dors Venabili.

I suppose none of Asimov’s laws specifically forbid kissing humans.

I need to go back and read Foundation.

My daughter came home with this approach to distance and got a 30 min diatribe about how foundational the Pythagorean theorem is to understanding the universe when she opined how dumb it was.

The Pythagorean Theorem is huge! It really is foundational for almost any understanding of distance.

You’ve extended my frontiers, I can think the theorem in the abstract now without losing touch with the ground, even up to 3D… you’ve made Maths fun…

I like that phrasing: “thinking in that abstract, without losing touch with the ground.” It’s what math should be!

Wish you were my maths teacher 🙂

Thanks–glad you enjoyed this approach!

Here: http://www.exploratorium.edu/cooking/meat/ 🙂

YES

Ben

Terrific post. This reminds me of a conversation at the Park Math blog called Death to the Distance Formula (IIRC)

I not only carry this up to 3 dimensions when I talk about it, but I also use this as a reminder of the fact that absolute value is defined as the square root of the square of a number when looking backward to the definition of distance on a number line. When they see the jump from 1 dimension to 2 dimensions, then the 3 D distance formula is easy peasy/

Hey, cool–I’d never thought about the 1D case! It’s really exactly the same.

More transparency for those formerly black boxes!!!

I really appeciated your post it was very engaging. I use the same approach to “discovering” the values of the unit circle. For pretty much the same reason – students do not have to memorize what they understand. 🙂

Definitely! The unit circle is a good example.

Also the fact that (sinx)^2 + (cosx)^2 = 1. Often accepted by students as a black box, but a trivial two-step proof if you pencil in a right triangle.

I’ve taken this approach too and want to offer a word of caution about abstracting to the formula too soon. Some students need to work with/ live with those specific cases for longer than others do. If I try to make the leap with them too soon even though I’ve tried to help them construct a “transparent box,” they end up scrapping it for the “black box” because “at least it works.” I think when I get to it this year I’ll build the box over several days… Do you have other thoughts or comments on becoming too abstract/generalizing too quickly? Of course the other piece is that some students are ready to generalize _very_ quickly, and building the box more slowly would be _very_ tedious for them. Maybe the answer is to toss them the 3D version to mull over too (although I suspect they’d devour that one as well)? Or maybe something about minimizing distances on a 3D object… (something like this: http://www.qbyte.org/puzzles/p006s.html).

Yeah, that’s a really good point. If you push students to make the leap to abstraction too fast, they’re liable to wind up extra confused, because they struggle to sift the conclusion (that the formula works) from the explanation (the proof of why). I’ve seen students believing they basically need to re-prove the formula every time they use it (which is fine in some cases, but unwieldy and slow in others).

Your idea of differentiating the instruction, with nice puzzles for the kids who finish quickly, is a good way to manage. Often there are interesting applications or extensions of a formula once you’ve proved it. (I’m looking forward to checking out that 3D problem you linked to!)

It’s also worth pointing out that the abstraction step is the essence of algebraic thinking. One of the ultimate goals is to help kids make those leaps of insight with more independence and confidence. For important formulas (key trig identities, definition of derivative, etc.), I’ll include test questions asking for a proof of the formula, to prompt them to review that abstractifying process, and help them build up some experience with it.

Yeah, that’s a really good point. If you push students to make the leap to abstraction too fast, they’re liable to wind up extra confused, because they struggle to sift the conclusion (that the formula works) from the explanation (the proof of why). I’ve seen students believing they basically need to re-prove the formula every time they use it (which is fine in some cases, but unwieldy and slow in others).

Your idea of differentiating the instruction, with nice puzzles for the kids who finish quickly, is a good way to manage. Often there are interesting applications or extensions of a formula once you’ve proved it. (I’m looking forward to checking out that 3D problem you linked to!)

It’s also worth pointing out that the abstraction step is the essence of algebraic thinking. One of the ultimate goals is to help kids make those leaps of insight with more independence and confidence. For important formulas (key trig identities, definition of derivative, etc.), I’ll include test questions asking for a proof of the formula, to prompt them to review that abstractifying process, and help them build up some experience with it.

Love the metaphor of black boxes! I long ago did away with the Distance Formula and replaced it with the Pythagorean Theorem. Another black box that I made transparent using the Pythagorean Theorem is creating an equation of a circle by using the definition of a circle…the set of all points (x,y) that are a given distance from a given center (a,b)…apply the Pythagorean Theorem…the hypotenuse is the radius.

I can’t wait to read your other posts. Love your “bad” drawings!

Elaine

Yes! So important with circles. The “standard form” of a circle’s equation is completely natural under this view… and arbitrary nonsense if you try to sidestep Pythagoras.

I like when you say “cold, mechanical, and not particularly romantic.”. I want romantic math 🙂

UC Berkeley Prof. named Frenkel apparently wrote a book called “Love and Math” or something like that–a good place to start if you’re looking for mathematical romance, I’d imagine!

I love this – my science class requires only a small amount of math, and what we cover is fairly simple algebra that my more well-prepared students can do in their heads. I use the opportunity to show the students how to derive their own general formulas that can be applied to more complex problems. It’s fun, and the 2-3 “Oh, wow” comments make it worth the 25 “Huh? What?” looks.

Interestingly – the power to derive my own equations by figuring out how I solved a simple example is something that mostly eluded me in math classes. I really picked it up when I was learning to solve math in college-level science classes… Word problems, for the win.

Well said–and there’s a chance that those even with those 25 “Huh? What?” students, you’re helping plant a seed of an idea that’ll grow later.

That ability to generalize from examples to formulas is one of the skills people usually forget acquiring once they’ve acquired it. It’s worth remembering how alien that logic feels at first brush.

Make sure you put that X= in front of the quadratic formula. I just took one point off your article. =) Nice job though in all seriousness – that’s exactly how I teach distance.

*tip of the hat*

Interesting connection to recipes and not knowing what the ingredients tasted like. The whole idea you are discussing reminded me of a blog post written by Michael Pershan entitled “Beg, Borrow, and Steal…”

I am in complete agreement on the need to not have black boxes in math!

Yeah, I liked that post of Michael’s! It was interesting seeing in the comments that he was reflecting more on the feeling of having a limited, black-box-y repertoire as a teacher than as a student.

Thanks for putting the ‘black box art’ up to engagingly capture our (math teachers’) attention. Another specific example of ‘black boxing’ is the potentially mind-killing slope formula in Algebra I. I stopped using it a few years ago — or rather, I started letting students find rate of change in simplistic real world scenarios without referencing the formula. Much better results — minds awake!

Good example! It wasn’t until my second or third time teaching precalculus I realized how much you need to emphasize slope as rate. Otherwise they see it as a sort of arbitrary, formula-defined thing, which leaves calculus as a pretty silly exercise.

You should write something about the Pythagorean Theorem, I’m sure it’d be plenty amusing.

I believe I have a total HATE against black boxes. If I don’t know, at least superficially, how something works, I cannot remember that. Just think that I cannot remember the systems for irrational disequations because I was absent when the teacher explained it, but I myself created a couple of theorems (I assume they already exist, too, but I played with them without knowing).

I knew how to calculate everything for a vector by pure logic before my teacher even started teaching that. I found it pretty obvious that applying the Pythagorean Theorem once on the first two vectors (unit vectors for x and y) and the result with the third vector (unit vector for Z) got me the coordinates of the full vector.

Mmm, I’ll keep that in mind about the Pythagorean Theorem. It might be fun to explore different proofs…

Your “build it for myself” approach reminds me of a lot of strong math students. When my dad (now a professor) was in school, he used to listen for the statement of a theorem, then stop paying attention to the lecturer, and just try to prove it on his own. He says it was great practice for him (though he doesn’t necessarily recommend it for everyone).

This is so weird reading this post. Yesterday I did some revision with a class I have only just met on applying Pythagoras’ Theorem. When I offered a problem involving finding the distance between two points, I was surprised to see the students grab for the recitation of a formula (incorrectly recalled BTW), even when I presented this as part of a set of application problems. It was like the formula overwhelmed all thinking. It made me realise that in my mind, I never remember the formula – I tap into my understanding of the underlying triangle – ableit very rapidly.

It was fun helping the class see how to access this thinking. I think a key to getting this happen was giving concrete values of points, insisting on them sketching the points (roughly) and drawing, or at least, discussing the triangle dimensions – and only in the last problem using variables. Which is *exactly* the sequence you showed in your post! 🙂 So was pleasantly weird reading this post – like seeing my whiteboard but drawn by you 🙂

Thanks for the great exposition!

Thanks for the story! It’s sort of funny that the two of us would converge on exactly the same explanation, but then again, the whole point of math is its universality. Still very weird, though, that you’d come across my exposition just a day after giving yours. Coincidences… 🙂

See my commentary on black boxes in mathematics in this blog piece: http://rationalmathed.blogspot.com/2007/09/looking-further-at-multiplication.html

(I realize that this is a late comment, but I just found your awesome blog.)

I’ve never used the same terms to describe it, but this is exactly I how like remembering/knowing things, especially math. My best math teacher always told me to focus less on the formula, but remember how to get it. I took that advice to heart, and as a result, I am now a “math person” who can’t remember most formulas off the top of my head. But I can easily figure them out for you in a few moments.

It makes me sad to realize that this really isn’t the standard method of teaching math. Especially to those students who obviously have trouble memorizing formulas. I tutor math, and it is amazing how often I see students have that “aha” moment of real understanding when I actually explain what a formula or method does, and why it does it.

Yeah, it always amazes me how many students think that the content of math is strictly limited to using formulas. Sure, using formulas is important, but so is their meaning and origin.

(And thanks for reading!)

Great article and pictures! I hope to point students to this website sometimes to make them more aware of their own “black box” thinking.

Another kind of black box, often literally so, is the student’s calculator. In my calculus classes I like to compare it to an ancient oracle or magic spell, because students have exactly the same kind of mind-set toward their calculator as people once had towards an oracle: you ask a question, it produces an answer from apparently nowhere, and the answer must be true because the black box said so (justified by the gods long ago and now presumably by some mysterious “smart” people who programmed it somehow).This kind of magical thinking leads to fallacies like assuming that the 10-digit decimal answer shown on the calculator is “exactly” correct, or (even worse) trusting a ridiculous answer that results from erroneous input without considering the reasonableness of the answer.

I want my students to think about how amazing it is that they have just “assumed” that their calculator can correctly produce the sine of an angle or the logarithm of a number without ever wondering how on Earth it could possibly do so. (Is there a little gremlin inside carefully drawing a right triangle and measuring the lengths of the sides?) Then when I develop the power series for the sine function or logarithm function it can be presented as finally stripping away some of the mystery and opacity of the black box.

I have always been antagonised by the black box, being told to leave assumptions at the departure gate, I always want to check the flight record, even when there isn’t a crash, to be confident that the black box was recording the right variables if there had been an adverse outcome.

Sadly, I found this doesn’t work for some people.

A friend of mine was having issues with the “special cases”, as my teacher calls them. (a+b)^2 = a^2 + 2ab + b^2 and (a+b)(a-b) = a^2 – b^2. She’s actually a smart lass, so I thought I could help her by telling her how it’s done. After all, it’s simple distributive property right?

As it turns out, after I told her, her answer was “How do you expect me to memorize all of this?”. And keep in mind, this is someone I once thought to be the type of person who would be able to figure this out in an instant.

I think that’s the main problem. The problem isn’t the black boxes themselves. It’s that the students acknowledge and, worse, don’t make an effort to look in them. They think of math as a memorization exercise, as opposed to a thinking one. They don’t realize that everything has a reason to be. If the teacher tells them a formula, then they believe it, even if proven wrong.

Another thing: my teacher has recently taught us these formulas, and did something I found clever: he SHOWED how to get to them. He demonstrated them with proof and knowledge we already had (even though some of it was unfounded, but I don’t blame him, because to get to such conclusions we’d need trigonometry way too advanced for 9th grade).

We had some premises, a goal, and had to reach a conclusion. He showed every step of it, and made sure everyone understood. In the end, he wrote down the formula. The kids wrote everything, including the formula.

Next day, when he asked a student to solve an exercise related to it, and keep in mind, this was a front row student: they saw the whole process unfold, they didn’t know anything. They didn’t know the formula, and they didn’t know how to get to it.

And honestly, that’s what I think is most lacking in kids’ education: it’s not that we need to tell them what’s in the black boxes: it’s that we need to tell them to look inside themselves.

As a tutor, I am frequently bothered by students who come in for math help, but when I try to explain how something works, just come back with “can’t I just use the *blah* formula?”, or if they don’t already have something memorized and handy, “isn’t there some formula I can use here?”. So, I know I’ve got someone very special when they see things my way, who can think of math as a toolbox of strategies, not formulas, who can explain their reasoning clearly and understandably, and can usually find the shortest route to a solution among many. I just wish I could have gotten to the rest of them years before, before they’d been infected by the “plug-n-chug/math is boring” virus.

When I teach that formula, we start by using Pyth. Th’m and then summarize it as

“d = sqrt[ (delta-x)^2 + (delta-y)^2 ] ”

to help connect the pieces of that formula with the numerator and denominator from slope.

Eventually I show them the formula with the subscripts as another way to write it, but I never require them to memorize that formula.

I thought this article was really interesting.

However, I think that there are still black boxes in math (sort of).

Let’s say we take a statement and prove it. Our proof relies on more statements, so we prove those as well. However, if we keep doing this, we run into one of three issues:

Definitions/Axioms. (Insert statement of Axiom of Choice here). This is an axiom, you cannot prove it. Just accept it as true (or 1 = {0} = {{}} and this is true because this is how 1 is defined in ZFC)

Regression: Proving statement a1 relies on the proof of statement a2, which relies on the proof of statement a3, … and this goes on forever

The circular argument: Proving statement a1 relies on the proof of statement a2, …, which relies on the proof of statement an, which relies on the proof of statement a1 (essentially x is true, so x is true but with more fluff thrown in)

Ooh, your comment strikes at something I’ve been thinking about this week – pretty good for a comment on an eight-year-old post!

In this post, I treat “proof” and “explanation” as roughly synonymous. To explain a mathematical truth is to prove it.

But today I see them as pretty different. Proof grounds a new statement in the existing mathematics; explanation grounds a new statement in the existing psychology of a person’s thoughts and understandings.

So, as you say, proof can’t ever unlock all of the black boxes; you can only trace everything back to axioms, and those are (on some level) arbitrary. But *explanation* can succeed in this way; you can trace statements back to physical intuitions or commonsense ideas that feel inescapably true to people (in a way that axioms may not).