I’ve been having a little argument with five-years-ago me. The question is this:

Five-years-ago me? Throw his drink in your face. He’d tell you that rote thinking is the bane of his working days, that deep understanding is the whole point of learning mathematics. He’d tell you: *No black boxes, ever*.

Today-me is less convinced. Don’t working mathematicians, from the ground floor all the way up to Andrew Wiles, sometimes use black boxes? Isn’t it common sense that sometimes you need to use tools that you can’t build for yourself?

I’m still wary of equipping students with black boxes, but these days I’m willing to do it, so long as three conditions are met. I hesitate to share this crude checklist, knowing my colleagues out there in the profession will have wiser ways to frame the tradeoffs. (After all, aren’t checklists too binary, too black-and-white, for an idea as elusive and shaded as “understanding”?)

Nevertheless, my checklist goes something like this:

When I taught my 17-year-olds the product rule for derivatives last year, I didn’t give them a proof. We talked through a few examples, and that was it.

“How do you think we’d prove this?” I asked later.

“Limit definition?” they said.

I nodded, and we left it at that. The proof I know is a clever algebraic trick; satisfying, but not terribly illuminating. I don’t really care whether students know the product rule’s origin story, so long as they know that it *has* an origin story.

By contrast, take my 11- year-old students as they begin secondary school. Many know lots of impressive “maths” (as they adorably say): they can divide fractions by fractions, subtract negatives, and state the n^{th} term of an arithmetic sequence.

And if you ask them to explain why a technique works, they just describe the technique again.

In their view, mathematical methods aren’t rooted in reason, emerging by natural processes of logic. They’re plastic flowers popping out of the pavement like magic. It’s not just that they don’t know why these methods work; they’re fundamentally unaware that “why” and “how” are different things.

To use a black box safely, a student needs to know there’s something they don’t know. If that isn’t happening, then I shun black boxes like I shun black bears.

Some techniques are not that enlightening—but you need them anyway.

I’m thinking of a three-act lesson where students estimate the number of pennies used to build a massive pyramid. Working from first principles, they can mentally dissect the pyramid, breaking it down into layers of various sizes. But once they’ve done that, they still won’t know how to total the number of pennies.

They need a formula: the one for the square pyramidal numbers.

Deriving it would be an impossible chore in the confines of a short lesson, and wouldn’t play to the learning goals. We’re left with two choices: (1) Deny students the formula, thereby forcing them through a long, tedious, repetitive computation, or (2) Supply students with the formula, a handy shortcut they don’t totally understand.

I’m comfortable choosing Door #2. After all, part of being a mathematician is tapping into the wisdom of those who came before.

This year, in an ambitious move, I tried to teach my 12-year-old students about square roots. In particular, I hoped they could learn to flexibly employ the rule √ab = √a√b, to simplify expressions like √300, or √72/√2 or √20 + √45 + √180.

In the immortal words of Rick Perry: Oops.

I pushed them too quickly into technique, and then watched them rehearse a rule they didn’t understand. All struggled; many rage-quit. They came to see square roots like an Old Testament plague. Luckily, there’s a simple solution:

*Don’t make them simplify square roots*.

They have no practical or intellectual need for this technique right now. They need to build numerical and geometric intuition about square roots first. No reason to thrust them into the deep end of this quasi-algebraic pool.

This is a surprisingly common tale in mathematics education. We rush headlong into technique, trying to outrun an imaginary time-monster. So I’m always reminding myself: *Be patient. Build context. Go concrete before you go abstract.*

To recap, I’m comfortable with students using a technique they can’t justify only if all three of these conditions are met:

Now the real question: when are these conditions met?

If you ask me: Almost never. Basically, it occurs when you’re teaching sophisticated students a piece of mathematics not for its own sake, but for its applications. Engineers, psychologists, and environmental scientists don’t necessarily need to trace the derivatives of sin(x) and cos(x) back to the squeeze theorem.

But I know this isn’t how most black boxes get deployed.

More often, it happens when your back is against the wall: Students arrive at your door unprepared for an immovable high-stakes exam. The shortcut to decent scores leads away from understanding. You face two repugnant paths: forsake the students’ learning to preserve their economic opportunities, or vice versa.

Many of us seek a middle way. We try to carry both treasures up the steep mountainside. But all too often, we arrive at the top to find that the learning is gone, vanished from our hands. We look back and see it scattered along the path. Step by step, we let it slip from our fingers, not even realizing.

To the teacher on that lonely mountaintop, I offer neither applause nor condemnation. Just sympathy.

I’ve been there.

I would say that I tend to avoid the black box, but there are sometimes when that is the best way to go.

I don’t need to understand the derivation on the cubic formula to know that there is a cubic formula and know how t apply it. I can trust Cardano. I know that I could, if I had the desire, read it carefully and follow the derivation, but I haven’t had the need.

I also know that the cubic equation is an important marker in the history of mathematics. It was through working on the cubic formula that we get the first use of complex numbers. I okay with complex numbers, and I see why complex numbers might arise when looking for the roots of a cubic equation, but I don’t know the specifics of their origin story.

And finally I know that Gallois proved that there is no general quintic equation, but I have really very little clue how is proof is constructed.

Physicists and engineers don’t really need to understand Cauchy’s definition of the limit to apply calculus. But they still know how to make calculus work for them.

There are places to make a decision to investigate the foundations, vs appreciate the structure, vs. jump straight into the applications. And after following one path, make the decision to circle back and investigate more deeply, or to head in completely new directions.

I love the way of thinking you’ve outlined here. I also look for connections or tools when opening the box. The pyramidal numbers I might want to have them practice how to ask Wolfram|Alpha to get the answer. The lesson about how to find those sums is probably a different lesson.

Ironically I’m writing a product rule lesson as I read this. I’m also omitting the proof. Not sure where there’s a need for that. I want them to see proofs as a way of knowing, but that’s not this lesson. But I do want them to make sense of the product rule. Remembering it like invert and multiply is better than nothing, maybe. But if they visualize the rectangle that’s better, and if they know why we can ignore the corner, that’s better yet.

Ok, let me throw out an area where I have embraced the black box… Annuity formulas. I can derive them, there’s some cute tricks involved OR I could let them us a TVM (Time value of money) solver. By letting them use a TVM solver, I can ask much more sophisticated queestions easily. Like “How much interest do I pay in the first year of my mortgage?” The formula is so complicated with parentheses that asking students (even college students) to do it correctly on a calculator is probably only about a 1 in 4 shot of being correct.

Funny, as one time mortgage trader, I know only one formula — the compound interest formula. Everything else falls out from that.

Net present values, future value, annuity payments, all variations off of the same formula. If I needed to work out the total interest paid in a year, I would derive it. I wouldn’t even know where to look it up.

But, when it comes to the yield to maturity calculation of a bond, there is no formula. Numerical approximation is the only way to do it. Add in optionality and Monte Carlo simulation is the only way to go. And once you cross that line, it truly is a black box.

Although the annuity formula is shunned as an ugly duckling, it has a hidden elegance: https://welltemperedspreadsheet.wordpress.com/2013/08/19/fast-formulas-1-new-insight-into-an-old-formula/

I simply wish someone would invent a method of time travel so you could have been my math teacher.

A-MEN!

Okay, I don’t know how it is for other people, and my thinking is really different from others, but personally I benefit a lot from knowing the logic behind techniques and hate using them without understanding. I almost always cannot go forward and use a technique I don’t fully understand. I can’t use it now and learn it later. Doesn’t work for me. It’s not only less fun, but also more difficult. For me at least. I’m having the same problem with my professor who tends to skip over the details of how and why something works and I just sit there lost and understanding nothing. Whereas when I understand the reasoning, I come up with all sorts of applications for it and use it efficiently. So I tend to agree with the five-years-ago you.

I do agree with your statement that understanding of the logic is highly beneficial. But sometimes high school or nearby Math does tend towards the easier side. You could open the black box and without much difficulty figure it out. But sometimes, a new piece of knowledge can itself get so overwhelming that the intricate circuitry that runs it, may fully throw you off.

For instance,the Binomial Theorem has a plethora of applications and various ways in which it can be manipulated. That is often treated as a black box, with many not knowing about its origins or inner workings. But once you get comfortable with its applications, you can get to the derivation quite easily. Even better, once you have its fundamental understanding, extending it to multinomials is a breeze.

Basically, along with all said by the Author, accepting the black box isn’t a problem as long as you retain the curiosity and urge to peek in at least once. Or taking your time with it, and then trying to get to its innards.

Hope it makes sense 🙂

Excellent discussion of math teaching (as usual). And to me this is one of the most important questions math teachers need to grapple with.

But I disagree with you about your first example, the product rule for derivatives. Should you teach them the proof? I agree with you: no, not enlightening, does not enhance understanding.

But it’s not hard to give an intuitive argument for why the rule makes sense. Not a proof, but something to make it meaningful and NOT a black box. Two ways:

(1) Use the area of a rectangle and consider A = L x W, new A = (L + delta-L) x (W + delta-W) … found in many calc textbooks. Nice visual argument.

(2) My way. Kick 2 students out of class for a made-up reason (but then tell them to stop just outside the door and watch). Give all 20 remaining students 8 M&Ms each. Total M&Ms = (M&Ms per student) x (# of students): T = M x S = 8 x 20 = 160. Call the other two students back in, bring all students up to 11 M&Ms each. So now T = 11 x 22 = 242. Is (delta-T) equal to (delta-M) x (delta-S)? 82=3×2? Wow — not even close. So how many new M&Ms did I hand out? Well, first I needed 3 for each of the original 20 students, then I needed to give the 2 new students the initial 8 each, and finally I needed to give the 2 new students 3 each: (delta-T) = (delta-M) x (S) + (M) x (delta-S) + (delta-M) x (delta-S). Then you need to see why in the limit you can ignore the last term … I’m not sure they buy that last part, but at least they can get an idea why this bizarre rule makes sense.

This approach applies intuitively to various applications, such as a company selling oranges, price is 70 cents and falling 2 cents per week, number sold is 1000 and increasing 60 oranges per week, what happens to revenue? Stewart’s textbook has some examples like this.

Evan Romer, Susquehanna Valley HS, Conklin NY (retired)

The problem with providing proofs at high school is twofold.

1) a large number in your class will be bored, either because they can’t follow it or because they are not interested to follow it. I don’t like boring my students. (It’s different at university – they signed up for it.)

2) opportunity cost. While providing a proof that is not required you are not teaching them some other skill that would be useful. That’s why I don’t teach differentiation from first principles — it’s a whole lesson lost that will never be recovered. And every lesson is precious.

I found this blog while doing some research for a college algebra course I’m taking and realized that almost all my algebraic understanding is “black box”-esque.

At first I didn’t understand the problem, but when I applied it to my own field (English) I realized the implications. I plan to go into teaching, and I wouldn’t want students to use rote memorization in lieu of genuine understanding of literary concepts.

This is some thought-provoking stuff…though I still may never understand the practical application of a logarithm. 🙂

In a similar vane. When is it ok to teach a technique without an application? This is something i allways struggled with. Disclosure: im an engineer.