Math under pressure

This outstanding TED Talk by Barnard’s president is mainly about choking under pressure. But how interesting that the example Professor Beilock spends most time on is girls’ learning math.

One of the excellent points she makes so well is that there’s a difference between knowing how to do something, and being able to do it when the pressure’s on.  And as you have probably experienced yourself, the pressure is in some sense always on.

I’ve experienced this since my school days, and I’ve done my share of studying this issue and experimenting with various best practices. When it comes to preparation for math tests of any kind, I consider this issue to be of equal importance to actually learning math.

I know. It sounds like heresy. But I know it’s right. So we use a three-pronged approach to preparing for math tests and math competitions alike:

  • Learn the necessary math to fluency
  • Identify and resolve all your performance/execution issues (per the above)
  • Strengthen your ability to critically deconstruct and to creatively synthesize

We give equal weight to these keys to success, because we understand that it isn’t just about what you know. It’s also about what you can do, and how you feel when you do it.


Goodbye, Princeton. I loved you while it lasted.

I will no longer be recommending Princeton Review’s Math 2 prep books, because their latest edition features a test that is a problem-by-problem parody of an official College Board test.

I don’t have a problem with close copies of official tests — on the contrary, that can be a very sensible strategy for creators of practice materials. However, my methods rely on a rich collection of problems that are different enough from one another that the student can come to generate underlying principles that apply broadly to many different kinds of problems.

Those methods of mine are undermined by problems that closely mimic other problems in the training corpus.

Therefore, I have to reject PR tests, because they follow a strategy that undermines mine.

(Alas, poor Yorick.)

Dealing with repeated challenges (part of “how to be a brighter student”)

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:

  1. Think about what you just did
  2. Think about what was helpful about it
  3. 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.

Expert level

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. Available on Amazon.

Work like an expert (part of “how to be a brighter student”)

February is the final date of the AMCbut it is also time to start your prep for 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?

If we do only what is required of us, we are slaves; the moment we do more, we are free.


Let’s pay attention to how learning the advanced stuff is different.

Some people never figure out this difference. They have some success learning the first lessons of a subject, and then they form a habit of “protecting” their knowledge of the basics against new information that could threaten what they “know.” (See Chapter 5, “Harnessing your mindset,” to remind yourself how this can happen.)

By contrast, you will instead maintain a growth mindset (also known as “beginner’s mind”), which means thinking of your knowledge as a way to get to expertise, rather than thinking of it as an accomplishment.

Concepts, tools, skills

When you’re doing it right, it goes like this: first you get the concepts, then you get the expert tools, then you develop skill with the tools.

For example, when you learn physics, one of the things you discover early on is the idea of “projectile motion,” i.e. the way things move when they are flying through the air. Once you understand this concept, you learn the equations that govern this behavior. These equations are the tools that let you predict exactly where a flying thing will be, and when.

But the really interesting learning is the part that is supposed to happen next. This is where you transition from knowing all the stuff, to understanding how to use it effectively.

Once you’ve gained those expert skills, word problems become not just doable, but obvious. You’ll watch baseball and see what it means for an outfielder to “be where the ball is going to land.” These are practical physics skills that you can learn after you know the basics, but they don’t happen automatically. You still have to work in order to get there.

A few examples

Let’s see how people do this in practice—both correctly and incorrectly—through a few general examples.

Consider people trying to learn math, especially math like, say, precalculus. They often spend their time memorizing formulas. But you now understand that although formulas are necessary for understanding math, they aren’t enough. Formulas are only tools. Once you’ve learned the formulas and read the explanations, you need to transition to learning the expert skills. In the case of math, this skill comes from solving problems. By trying (and, usually, failing a lot), people learn a lot more useful math a lot more

What about people trying to learn a language? They usually start with memorizing vocabulary. Of course, you have to know what the words mean before you can start putting them together in any useful way. (This is the basic concept of learning a foreign language: as Steve Martin said, “They have a different word for everything!”). The most common mistake here is to try to memorize all the vocabulary at the beginning. What’s smarter is to get a small collection of vocabulary words down cold, then start reading, speaking, and writing in the target language using that implied vocabulary. By doing this you work the necessary skills (reading, speaking, writing) that are built on the fundamental tools (meanings of words), as soon as possible.

Sports also have their drills that lead to mastery of the basic patterns of movement and attention required for success (as well as general conditioning). Once you have these tools, you can (and should) immediately start building the skills needed to win at the sport.

How we usually go wrong…and how to do better

Each subject has its fundamental knowledge that must be completely committed to memory before mastery becomes possible. But each subject will also allow you to keep working on the fundamentals instead of graduating to the skills of mastery, if you’re not careful.

That’s why it’s important not to confuse the fundamentals with the skills that are meant to be built on top of the fundamentals.

Why would we keep working on fundamentals? The answer’s simple. It’s because it’s always comfortable to work at something you’re already good at. It’s harder to work at something new. So, if you want to master a subject, there comes a point at which you need to decide that you’ve mastered the necessary core knowledge, then tear yourself away from working at the fundamentals, and start working at a new, higher level.

And this is why many successful language-acquisition programs insist that students begin speaking as soon as they have a minimum vocabulary. It’s also why math is best learned by solving problems rather than by reading explanations. (And it’s why no one’s ever gotten good at a sport by watching it from the bleachers.)

As soon as you can, you have to transition from learning more fundamentals to practicing. And that means learning from your mistakes.

Deliberate practice

Anders Ericsson (mentioned earlier, in Chapter 3, “10,000 hours?”) describes the idea of deliberate practice – an effort to systematically practice just beyond one’s skill level – as the key to accelerated improvement in any discipline. This sort of deliberate practice is difficult and demoralizing, as it involves constant failure. And it can be exhausting, because it challenges your mind at multiple levels at once. Not only are you grappling with uncomfortable new processes and techniques, but you are also monitoring your own process, and devising new challenges for yourself on the fly. So, it’s hard. But it works like nothing else.

In a test-preparation context, this means that you must begin taking practice tests as soon as you can reasonably expect to be able to answer some of the questions asked. As you struggle with the material, you will not only learn which bits of fundamental knowledge you are missing; you’ll also learn whether your pace is correct, whether your nerves are hurting or helping you, at what time of day you do your best work, how much sleep you need, and on and on and on.

With each mistake, you have an opportunity to be completely honest about the reasons for your error, which in turn leads to ideas about how to correct future errors by changing not only your knowledge, but also your habits and even your outlook.

Obviously, this is not the same as just doing an activity. For example, playing a sport for fun for eight hours a day is not the same as deliberate practice; deliberate practice would involve setting up drills to specifically focus on deficiencies in one’s skill and practicing those drills until the deficiency is corrected.

Ericsson draws the conclusion that natural talent means nothing without deliberate practice. In fact, in situations where we think of somebody having natural talent, an investigation shows that they just got started on their deliberate practice earlier than most.

When you want to get good at something, aim to do it the way masters do it. Then, when you fail, be gentle and honest with yourself. Look for exactly what you did wrong, observing yourself without judgment. Then address the mistakes, whatever they may turn out to be.


Mastery of the basics is just the beginning.

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. Available on Amazon.

Say NO to ROPO (in private education)

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).

What’s wrong with math education

“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.”