January is AMC crunch time. Later will come SAT Subject Tests and AP’s. Examine your test prep strategy. How will your hard work pay off, not only at test time, but also later in life? How will you utilize the skills you’ve refined over the course of your preparation to create a better you?
The mechanic who would perfect his work must first sharpen his tools. – Confucius
Now, let’s shift our focus from the creation of future you to the tools we will give him or her. I’d like to focus specifically on the theme of repeated challenge, since that’s what the future version of you is going to be grappling with. From our point of view, we might even say that that is the point of future you: he or she is going to handle challenges like your current challenges, only better. So let’s see how “better handling” of repeated challenges actually works.
How most of us think about repeated challenge
Most of us believe, incorrectly but stubbornly, that after we’ve overcome a challenge once, we will automatically overcome all future similar-looking challenges, as illustrated by this story:
Henry has done most of his homework for a class, but the last question is of a type he doesn’t think he has seen before. He tries a few methods suggested by the current chapter, but nothing seems to work. He feels a little anxious, but he decides to give it a break for dinner. After dinner, he goes back to your desk, and the solution hits him! He finishes the question and finishes his homework, and closes his notebook triumphantly.
Next week, he gets stuck again while working on his homework. Again it’s a type of problem he doesn’t think he has seen before, just like last week. And he figures that he’ll probably solve it quickly and easily, just because he solved it quickly and easily last time.
Note the mistake in the story Henry tells himself: it wasn’t quick and easy the first time, and it won’t be quick and easy the second time either. But if we are very careful to keep track of the details of how we solved the problem the first time, our chance of success the second time is much higher. Over time, this repeated process will become quicker, and it will seem easier and easier. But only over time, and over many repetitions, and probably with some mistakes and failures mixed in.
How most of us deal with repeated challenge
After overcoming a challenge, we move on immediately, and expect that we will be able to recall any important parts of the solution later. For example:
Grace has done most of her homework for a class, but the last question is of a type she doesn’t think she has seen before. She tries a few methods suggested by the current chapter, but nothing seems to work. She feels a little anxious, but she decides to give it a break for dinner. Just as she’s leaving her room, the solution hits her, based on an obscure method from a previous chapter. She goes back to your desk, finishes the question and finishes her homework, and closes her notebook triumphantly.
Next week, she gets stuck again while working on her homework. She thinks about the problem for a few moments, and no ideas come to mind. But then, she thinks back to the last time she had a mystery problem. She remembers that the solution came to her when she decided to break for dinner. So she decides to do the same thing this time.
But it doesn’t seem to work this time. She finishes dinner, returns to her desk, and still there is no solution. There must be something she did last time that worked, but she just can’t remember all the details. She’s stuck. That’s funny, she thinks. It seemed so obvious at the end last time.
How to better handle repeated challenge
If you want to handle repeated challenge in the best way, you have to start by realizing that the goal isn’t to change your challenges. The goal is to improve your ability to handle them. This takes an extra step or two that we’re not used to: reflecting on current successes just after they happen, and leaving notes for your future self to benefit from. Here’s how that looks in practice:
Rusty has done most of his homework for a class, but the last question is of a type he doesn’t think he’s seen before. He tries a few methods suggested by the current chapter, but nothing seems to work. He feels a little anxious, but he decides to give it a break for dinner. Just as he’s leaving his room, the solution hits him, based on an obscure method from a previous chapter. He goes back to his desk, finishes the question and finishes his homework, and closes his notebook triumphantly.
Then, thinking forward to “future Rusty” and the challenges he will have to overcome, he opens his notebook again and spends a few minutes writing down what he just discovered. It comes back in slow motion, and he gets it all down: the feeling of being stuck (so future Rusty can recognize it for what it is more easily later), the ideas he considered and rejected (so future Rusty can get better at analyzing options), the decision to break for dinner (so future Rusty can learn from his lucky experiment of solving a problem by giving it some space), and the flash of insight itself (which, Rusty now realizes, actually came from a mental survey of cryptic hints the teacher had given when assigning the homework). Now he has it all down.
Next week, Rusty gets stuck again while working on his homework. He thinks about the problem for a few moments, and no ideas come to mind. But then, he remembers that he had this feeling last week. He turns back in his notebook, and reads the notes he left himself a week ago. Suddenly it’s much clearer. He goes through the current problem step by step; he reviews what the teacher has said this week; he re-solves last week’s problem. He still can’t find the answer, but he doesn’t worry about that. Instead, he breaks for dinner. He’s pretty sure he’ll figure it out, even though he doesn’t yet know what the solution will be. Sure enough, while Rusty is eating, he thinks of something that might work. When he gets back to his desk, he works out the entire idea. It works! He breathes a small sigh of relief.
Okay, so what are the steps again?
Whenever you solve a problem that you think you might face again, think forward to what will happen when you are confronted with a similar problem in the future. That will give yourself the idea of what to do this time, so that you will be able to take advantage in the future of what you learned just now. So:
- Think about what you just did
- Think about what was helpful about it
- Write a note to your future self
Don’t skip that third step! Writing that note to your future self means you don’t have to rely on your (let’s face it, imperfect) memory. This habit is a bit like being a time traveler, in a way: once you ask yourself what your future self would want you to do right now, you’ll find yourself taking actions that set you up for huge successes. With practice, you’ll get these successes again and again. In this way you can think of yourself as a team of you’s: past you’s, current you, and future you’s, all working together to shape the path to best fit the team (i.e. to best fit you).
If you want to get good at something over time, you have to analyze your performance. “Reps” alone won’t do it.
You will face different kinds of repeated challenges in the future. Not just tests and courses, but interviews and jobs, and difficult conversations, and planning for a career and family, and beyond. The same tools apply.
Would you like to read more?
This post is an excerpt from my new book, “How to Be a Bright(er) Student: The Craft of Developing Your Brilliance”, a step-by-step guide to unlocking your inner potential and become the math whiz you were always meant to be. Soon to be available on Amazon.
Return On Planned Obsolescence (ROPO), I have recently learned, is basically the extra money Apple gets when people buy a new phone while their old still works perfectly fine, but just doesn’t have the newest bells and whistles. It can also refer to extra money netted by a manufacturer who purposely shortens the lifespan on a product in order to encourage more purchases faster. (If you are as new to this idea as I was, you might enjoy skimming this, this, or this.)
When I saw this acronym for the first time, it struck me that this idea is built into the tutoring industry, and that that’s a real problem. Now, I’m not talking about the version of ROPO where you make a disposable thing so people will buy more. The tutoring analogy of that would be giving students less-than-great help in order that they’ll need more.
To be clear, I don’t seriously think anyone’s deliberately doing that. But I am talking about quality control in a broader sense.
The fact is that it’s pretty profitable to have students — or better, for your tutors to have students — who keep coming back, week after week, for help. Every tutor has to eventually face the question “what if I help this kid enough that s/he doesn’t need me anymore?”
For me, the really interesting part is that I’ve seen a few different sides of this question now: I have been the tutor, I strongly suspect I have been the student, I have been the tutor manager, and I have been the finance guy looking at the firm’s metrics.
And I am here to report that this idea is more important than I ever gave it credit for. I’ll explain why, but first, two more bits of context:
Recurring revenue is a very, very good thing for a business to have for many reasons that have been hashed to death in the blogosphere, such as here, here, and here. (Cue entrance, Captain Obvious!)
Owning a tutoring business often feels like serving two masters: you are an educator, and you are a business person. At least that’s how it can feel on a bad day. And if those two masters disagree on the best course of action in just about any situation, you as are in serious trouble, because you are likely to regret whatever you do next.
My big takeaway is that you can and should choose to be an educator first, which means that every single tutoring session with a student should feel to the tutor like the last. You aren’t ever doing the same old thing. You aren’t “working on an assignment.” You aren’t “making progress.” You are solving a problem, removing a barrier, addressing an issue. You are finishing, finishing, finishing.
You are fighting against recurring revenue, every single time. And suddenly, with that realization, comes an understanding of a choice that matters to me, and how I can move even farther in that direction: buck the trend, and keep finishing.
Let me try to illustrate why that isn’t actually bad business, even though it might sound that way:
I had a session with a remarkable student a few years ago. She was having trouble with a math class — well, really, with a math teacher. The teacher and the student had different expectations on a few levels, and neither “spoke the other’s language.” I felt I understood the student’s position and the teacher’s position pretty well, so I set out to teach the student how to see the class through the teacher’s eyes. It led to an energetic and enthusiastic high-level discussion in which the student originated a number of spot-on ideas for how to better give the teacher what she wanted, without watering down her own experience in any way. From the outside, I can’t imagine anyone would have identified that session as “tutoring.” It was just what that student most needed.
The student never returned for math help, and when I later asked why, the answer was simple: she no longer needed it. Now, how much recurring revenue did I turn away that day? Answer: don’t think about it. It’s not the point. That kid is going to do great things; it was right and good of me to help her with no thought of holding back. (If this doesn’t seem self-evident, then you may be interested to know that her father has been an enthusiastic referral source for years now, because, luckily for me, these folks are not only smart but also conscientious and mindfully supportive. And while that doesn’t happen every time, it does happen surprisingly often.)
Here’s the lesson for educators: don’t save a “big reveal” for next time. Don’t ever turn on the auto-pilot. Don’t be complacent. If you really want your education practice to last, you have to be a better educator tomorrow than you were yesterday. And that means giving it 100% (no matter what the business coaches may say).
“Math punk” Tom Henderson has written a brilliant essay that I have co-opted and edited here.
In a nutshell, what is the problem with math education in the US? It’s that students are mostly trying to minimize feeling stupid rather than trying to maximize their ability to solve problems.
This manifests as “show me The Steps.”
Many students want a sequence of steps that they can perform that will give them an answer. This is not unreasonable; they know that their performance on exams, and therefore their performance on the All-Important Grade Point Average, is largely determined by being able to Do The Steps. So they want to know the formulas, so that they can float them on top of their short-term memory, ace the exam, and then skim them off.
For their entire mathematical careers, math has been a sequence of Steps, and if they get them wrong, they get red pen, bad grades, No No No Look What You Did. Plus, bonus, there is no apparent relevance of these algorithms other than To Get The Answer.
But that’s crap. “The Steps” aren’t math, and what’s more, The Steps aren’t generally useful in life. What’s useful is the ability to deconstruct thorny problems and figure out a way to tackle each of the pieces.
The Steps are seeing the sorts of symbols that count as “right”, and trying to replicate that dance of steps. It turns out that the easiest thing in the world is to look at a student’s work, and tell the difference between “Knows what’s going on, made mistakes and dozed off” vs. “Can memorize steps, has no idea what’s going on.”
Now, a better way to explain mathematics sort of looks crazy at first. It’s handwaving. It’s referring to certain groupings of symbols as “alphabet soup” and writing it down as a wild scribble with one or two symbols around it. It looks nothing like standard “math class” from the outside.
That’s because the better way avoids showing The Steps and instead shows enough of The Idea that the student can reconstruct what the steps MUST be.
And that brings us to a better way to learn mathematics: you get a fear-free zone, you check your ego at the door, you try a bunch of things that will wind up not working, you ask a pile of dumb questions, and before long, you figure out some crazy way to get the problem solved. And only then do you realize that your crazy, lame-brained, that-can’t-possibly-work solution is in point of fact the official method nine times out of ten. Because math is, at its core, just a collection of “the best way we could figure it out” stories, organized semi-sensibly, and with a specialized vocabulary and language on top.
So, what’s wrong with math education in the US? What’s wrong is: whatever it is that makes students uninterested in learning any more math than is required to minimize feeling stupid. My solution? Provide that safe space; find the genius in every question; and provide interesting problems to solve.
Ultimately, that’s all it takes to get students to say “Oh. That’s not nearly as hard as I thought it was going to be.”
Brian Greene makes the point about science, and it holds just as true for math: “We rob science education of life when we focus solely on results and seek to train students to solve problems and recite facts without a commensurate emphasis on transporting them out beyond the stars.”
Interested in communicating with a faraway friend without allowing anyone to eavesdrop on you? Of course you are; this problem affects a middle-schooler’s daily life, and yet it is also the basis for modern commerce: communicating your credit card number over the Net without allowing a thief to eavesdrop is a non-negotiable requirement for our economy. How is this problem solved? First, you try for a while. Then let’s talk about prime numbers and see what they can do for us.
Interested in taking a rocket to the moon? Well, if you want to drive a spaceship, you probably ought to understand how gravity works differently from your intuition when you’re far from Earth. Let’s talk about ellipses, and while we’re at it, let’s predict the next approach of Halley’s comet. (And if you’re a high-school sophomore or so, we can talk about inverse-square laws while we’re at it.)
What if we picked up a signal for an alien civilization on our radio telescopes tomorrow? Do you think they’d speak English? Not a chance. Let’s think about how we would communicate without a shared language, a shared culture, even a shared planet? Well, there’s one thing we have in common with any race advanced enough to send us a signal like that: they know math. Let’s think about how that might work…
One of the biggest rarely-asked questions in math is simply “why should I care?”
I love answering that one.
Well, I really mean to answer “what’s wrong with using the results of students’ standardized tests to evaluate teacher performance?”
On its surface, nothing. Indeed, student performance should be the primary metric of teacher quality. When I worked for Bodsat Prep, I was gauged by that metric myself: what matters is the point gain.
The main problem, though, is that the standardized tests don’t test everything that we want our students to know. In the same way that the standardized test at the DMV doesn’t tell you whether you’re a good driver (though it does catch some really bad drivers), standardized testing for schools doesn’t tell you whether you’re on track to be a productive member of society (though it does catch a number of people who really aren’t).
Also, if a standardized test samples say a random 10% of what one should know, then how well one does on such a test is, statistically, a good indicator of how much of the target material one knows. But if it’s a non-random 10%, then you can count on prepared students to know that 10% very well. And possibly nothing else.
The thing I can’t get out of my head when I read this NYT article on kids paying other kids to take the SAT for them is simply this:
All they charged was $3600? Cripes, that’s a bargain at ten times the price.
Don’t get me wrong: you shouldn’t cheat, both because it’s wrong (which should be enough reason) and tactically too risky (in case the first argument wasn’t enough).
But I mean, come on, let’s do the math: a one-percent increase in salary over your life is easily a five-digit number even if you’re kind of a slacker. Two significantly different SAT scores mean admission to schools of two significantly different calibers. And I doubt the salary increase we’re talking about here is just 1%.
For those of you who are fans of the Drake equation, which uses best-guesses to try to figure out whether there’s intelligent life out there, I challenge you to apply this reasoning to SAT prep.
In fact, you might even try to create an analogous equation governing this stuff, like I just did. (I hope you have more luck than I did; if so, please let me know.) But, equations aside, it’s not really that hard to think about.
To figure out what a higher SAT score is worth, just do the following steps:
First, get a lifetime earnings calculator. (Google it; there are many.)
Then, use it to estimate the student’s lifetime earnings, given that he or she attends the best school to which he or she can gain admission given the initial SAT scores.
Then, take the average (expected) gain in SAT scores given a particular preparation method.
Then, use the calculator to estimate the student’s lifetime earnings, given that he or she attends the best school to which he or she can gain admission given the final (expected) SAT scores.
The difference between the two lifetime earnings is the value of the higher SAT score.
And now that I’ve said all that out loud, I’m starting to realize that “only four-digit” prices for SAT prep only make sense for providers who can offer only single-percentage-point gains with a high variance, as delineated in this article in the Wall Street Journal regarding the average benefit of SAT prep.
Now these days, I am no longer doing SAT prep, having left Bodsat Prep in 2016. However, the work I do preparing students for the AMC competitions (as well as the Math 2 and Physics SAT Subject Tests) still seems to be governed by this math.
Since we don’t often see prices like this, the conclusion I come to is: almost no one is delivering reliable results. (Or the people who are aren’t also good at pricing.) Interesting.