Not just about the math

Paul Lockhart has a new book out.  The title (Measurement) will likely strike dread in the hearts of those who despise mathematics for one reason or another, but I mention the event not for the math but for the potential contagion of Lockhart’s delight in his work. He’s made a short video to accompany the release of Measurement. Even if you want nothing to do with the math, you may find that the way Lockhart is about it inspires and reinvigorates – that he reminds you of what you care most about and why you care about it.

Also, though, if you are looking for a gentle invitation into the world of numbers, shapes, patterns, and mathematical happening and inquiry (whether for your own sake or that of your young), you won’t likely find anyone more eager to hold the door open for you than Lockhart.  He knows we won’t all love math the way he does, but he sees room in it for everyone who comes inquiring after it.  No matter how many times they’ve been otherwise told they’re not welcome and don’t have what it takes.


Vitalization projects

A few weeks ago I had cause to revisit Anne Lamott’s Bird by Bird.  On the second-to-last page, she quotes a friend who likes to ask “How alive are you willing to be?”

It happened that the morning before, I was up against the edges of my own ease, a beginner among masters at something I’ve always wanted to learn. More than once I’d had to decide whether the possibility of success, which would offer me an experience of being alive and fulfilled that other things, more comfortable and convenient things, might not, was worth the discomfort and inconvenience afforded by my beginner status.

So the question struck me with some force, and after pondering it for myself, it occurred to me that it might prompt a powerful inquiry about our standards for children’s lives. We often find ourselves at odds with children, with the pastimes they are drawn to, the things they want to do with their time and energies and intellects.  We tell them and we tell ourselves that it’s for their own good that we seek to bend them this way, toward the pursuits that we think will take them where we hope they’ll go.

In the name of education and preparation, we enforce time spent on the things tradition has taught us to value, at the expense of the things that bring kids most readily to life – the things that bring out the brightness, the determination, the patience to keep at something even when it’s hard. We know our intentions are good, but is it getting us what we actually want for kids?

How alive are we willing to let them be?


The other day I wrote about waiting to offer help until it’s clear that help is needed and/or wanted.  Here’s a bit of follow-up to that:

When we skip the inquiry into the usefulness or necessity of a particular task or activity, we deny ourselves the opportunity to see what might be trying to make itself known in the space created by not pursuing that thing just then.  Any time we’re doing one thing, we have to be not doing another thing; everything we ever do requires the choice to not do something else. And it may in fact be that reading, for example, is not the most important thing for every child to be doing right away.  It may be that for some people the most important first thing to master is listening, that for others it’s building or sorting or observing or climbing or strategizing or swimming or questioning.

It’s possible that obstacles not only offer us, as Randy Pausch suggested, the chance to find out how driven and committed we are about something, but that they also give us an opportunity to find out if there’s something else that might be even better for us, at least right then.

What if, as a result of being inhibited in a particular way, we become able to find our way to the most productive, fulfilling, and otherwise beneficial pursuits available to each of us? The likes of which we might not have found if all of our attention was on that early reading, on getting that arm to move that one way? We don’t have to give up the difficult things to find this out.  They’ll be there waiting for us, inquiry or no.  We just might find a richness in the experience available on the other side of this inquiry that makes the difficult things easier to tackle; more fulfilling and useful against a backdrop of other tremendous purpose or reward because we’ve been willing to go looking beyond the confines of traditional mandate. We might also find that there are things we’ve struggled and struggled with, always held as essential, that turn out to be otherwise and that when we let our grip on them loosen, we get stronger, clearer, happier, healthier for what we become available to do instead.

Over the walls


There’s that great story about the patient who says to his doctor “My arm hurts when I do this,” hoping of course for a diagnosis or prescription related to his ailment. But the doctor only says “Then don’t do that.”

It’s a joke, sort of, about human behavior, pathology, and medicine in general.  It has some potentially interesting implications for learning, too, and what we do when kids don’t perform the way we want them to, or the way we think they should, when we think they should. I’ve been imagining an expanded version of this doctor/patient joke.  Of course in a real situation we would hope that the doctor would have more than simple functionality in mind, but here’s my alternate idea of how it might go:

Patient: “My arm hurts when I do this.”

Doctor: “Interesting.  What were you doing when you discovered that?”

Patient: “I was ______ [some activity or task].”

Doctor: “I see. Is that something you will need or want to do again? Would it be a problem if you couldn’t do it?

From here, the conversation proceeds differently depending on how the patient responds.  If this thing that he can’t do with his arm is something he doesn’t have much use for, then the advice from the original story might be sufficiently sound:

Patient: “Actually, no. It won’t really get in the way if I can’t do it.”

Doctor: “OK.  Then stop doing that with your arm.”

If, on the other hand (pun partially intended), the task in question is something the patient does need or want to be able to do, then more of a problem-solving approach would be called for:

Patient: “Yes. I need to be able to ______.  If I can’t then it’ll mean _____.”

Doctor: “OK.  Then let’s see if we can figure out a way for you to do it.”

The doctor is of course ready to offer her expertise, to do the diagnostic work it will take to get to the bottom of the arm’s limitation.  She’s ready to intervene to restore or create the functionality the patient is looking for. But she waits to find out whether or not the patient actually wants and needs that intervention before she embarks upon it.


We often skip over this kind of inquiry with kids. When we see that a child is not meeting a mark we’ve set, we turn quickly to the work of getting the child to meet that mark.  We make it a problem, whether or not kids ask for help with it.

We do this with the noblest of intentions, and we have our reasons.  The big one is that we’ve decided that certain things will get in the way if kids don’t learn how to do them on schedule.  I don’t happen to think we’re right about this, much of the time, because so far we haven’t updated our list of learning necessities as the demands of job markets, economies, and work have changed with time.  (Also, specific skills and bodies of knowledge don’t serve all people the same way.) We keep trusting that the things we’ve always insisted upon to equip and prepare young people for their lives are still and always the ones they need, and need first.  

There’s so much to gain in granting children the dignity of an exploratory type of conversation like this one between the doctor and patient.  By asking ourselves and asking kids whether an inability to do something, or a challenge that arises in the course of learning something, is actually getting in the way or might actually get in the way at some point in the future, we make it possible to see what is truly so and what more there is to see.  When we jump over this step, it removes kids from the process of navigating their lives.  We dole out “help” they haven’t identified a need for and haven’t asked for. And in making that choice we give them an experience of powerlessness and render the help itself difficult to receive and absorb.

Brick Walls

It’s of course true that getting that arm to move that way, or getting a young person to read early in life, or getting any other outcome we want may indeed be just the thing for the person in question.  I’m reminded of Randy Pausch’s point from his Last Lecture about the obstacles one encounters in pursuit of a childhood dream: “The brick walls are not there to keep us out.  The brick walls are there to give us a chance to show how badly we want something.” For some, the fussy arm or the difficulty reading turns out to be a brick wall.  It lets the person find out that this thing is worth all it takes to accomplish. For others, though, it might actually be a distraction from other more meaningful or beneficial pursuits.  At least right then, maybe always.

And then what about if there’s really something wrong with the arm or with the reading, something that isn’t about choice or will or commitment?

To me, this possibility is actually the most compelling reason to be sure that a person gets the benefit of starting from a place of his or her own clarity and commitment.  Anyone who has ever had the experience of a brick wall offered up by neurological wiring or other physiology (as with any brick wall) knows that it takes the power of personal, specific motivation to do whatever it takes to get up and over that wall.  Overcoming an obstacle just because someone else thinks you should – because someone else threatens or otherwise insists – is a much, much taller order.

Navigating that pre-mathless world…

My last post prompted this question from a reader:

…I am half way through the Mathematician’s Lament and am totally, utterly passionately sold. But…now what? I’m not a mathematician and fall into the “duh” populace of math paralysis. Who has a curriculum, a study guide, activities prepared to those of us who want to give this gift of wonder to our kids and allow the +,-, X,/ come later, and and naturally? I don’t know where to begin or what to show. I wouldn’t have known the triange in the rectangle thing if I hadn’t just read it in the book. So where do we find sources? (I intend to track down the author and ask him the same).:)

Here’s my response:

If you haven’t already, I’d recommend you have a look at Beyond Facts and Flashcards (read more about the book here); whether or not the mathematical content is right for your son where he’s at now, the book may be able you at some ease about your own ability to guide him through what he’ll need in the way of math.

The other thing I often recommend is that people who feel about all this as you do (“the duh populace”) start by looking around to find out what they’ve been using in the way of practical math all along, more or less without disaster, humiliation, or other unfortunate incident.  It’s there!  You’ll likely find out that you’re more savvy and capable than you’ve been led to believe, and that you have more to offer than you think when it comes to passing on useful stuff to your child.

Then when it comes to the other realm that Paul Lockhart’s talking about – the stuff that’s about finding beauty and wonder in shapes and relationships and numbers – you can go exploring.  The thing about mathematics as a pleasurable pursuit is that it didn’t exist until people started talking about what they noticed, and then started creating language for it so they could talk about it and share in the exploration and creation.  So the thing to do if you’re interested in finding out what’s in this realm of mathematical beauty is to go looking for the people who have found it fascinating and then see if there’s overlap of interest and intrigue.  You can Google things like “mathematics and beauty” and “interesting math discoveries.”

You don’t (nor does your child) have to love it, or even care at all about it in order to find your way quite effectively and peacefully through the world!  Sure, that thing about the triangle in the rectangle can be useful in the context of a math class, and can help facilitate the acquisition of subsequent knowledge, but unless it’s inherently interesting how shapes interact with each other, it’s likely to have at most cursory usefulness (and to try to get a kid to dig deep into math when it’s just plain not that intriguing to him or her is to make it more difficult to just learn it to the extent that it may be necessary in the context of preparing for a test or getting through a class; the more we let things be how they are the more smoothly things go).  Like anything else.  I’ve worked on problems like the triangle/rectangle one with kids who are fascinated by it and want to spend scads of time thinking about it, and others who shrug and say things like “Yeah, I guess that’s cool.” I don’t think that’s entirely the fault of the delivery.  I think it’s at least in part attributable to the spectrum of human preference.

The fact that we make a big production about math is not an indication that it’s any more important or essential (and I’m still talking about the big fascinating puzzle math, not the managing finances) as anything else, like art history or needlepoint, things that we’re perfectly fine with taking an interest in or not taking an interest in.  You can go after the interesting math the same way you would anything else, by asking people you know if they know things about it, or know anyone who does, and then asking those people what they read first, or where they started, what they’ve seen online about it that they like, etc.  Some of the stuff you come across will be over your head at first, but some of it won’t, and that’ll be where you start.  If you find it interesting.  If you don’t, then maybe, slowly, you realize that it doesn’t mean anything about you that it’s not interesting to you, just as it doesn’t mean anything about you that you’re not interested in needlepoint or engine repair. And you can choose to keep exploring, or stop!

Thanks for asking the question out loud, and for being willing to venture out beyond what you’ve been taught to believe about math!

Uniqueness is messy.

This American Life’s recent episode on middle school mentions Maria Montessori’s belief that the appropriate environment for a child of middle school age is a farm school.

What I’ve read about this idea and many other Montessori ideas sounds wonderful: young people at work and play alongside respectful adults who can teach them to do things as well as to know things, to apply their knowledge, to be physically active in the course of it… the list of desirables goes on.

But this and every other idea that seem delicious in the abstract start to feel a little less solid when mapped onto actual specific children.  Whether or not they could benefit from the activities proposed, whether or not the values instilled would be useful to them, all people are not equally available for all endeavors and pursuits.  And the timing of one person’s availability for something, their ability to receive and absorb it, is not likely to be the same as everyone else’s.  But that’s what we expect.  We want children to be unique, but at the same time, we’re wary of specialization.  It’s as though we want them to be unique sometimes, or later.  Not when it gets in the way.  If they get too interested in something too early, we tell them they’re not well-rounded.  We want a checklist that will be the same for all of them, of things they should all do first.  When that’s done, we expect them to be ready to be unique and to distinguish themselves. Of course, it often doesn’t happen that way.  Uniqueness and individuality is a tough thing to temporarily shelve.  Kids’ actual uniqueness, which is a messy, unpredictable, impossible to control thing, confounds us. Come to think of it, so is our own.

I think this is why it’s so hard to see the potential that kids already have, the capacities that are already in development when they’re 3, 7, 11 years old.  We might have trouble, for example, seeing that all of the time they spend with their Legos is building skill they might need for engineering, because we’re think that first they need to sit still and add columns of numbers; engineering is for later.  We might have trouble seeing that they’re learning how to think deeply and analytically because we think that first they need to be practicing their spelling; philosophical inquiry is for later.

We know that what we’re doing is not working, not making the most of kids’ potential, but we’re so fixed in this belief that we have to put the many through the one thing (even though it so often doesn’t come out right and is a struggle the whole way) that we can’t see all the things we’re shutting down that could launch individual kids on paths that are well-suited to them and their specific capacities. If the outcomes of educational efforts are to shift in any meaningful way, it’ll be because we confront our biases about what has to happen when, and our attachment to giving everyone the same thing at the same time.

And as we’re retraining ourselves to look at what’s actually there and what kids might do with it – the evidence of potential that may not fall into the traditional, recognizable categories – we’ll also need to stop laughing amongst ourselves at kids’ resistance.  As long as we’re behaving toward children as though the discomfort that drives their resistance is funny or cute, they’ll keep it up.  Because that resistance is not the personal attack on parents and other adults that we treat it as. Young peoples’ defiance is a plea – a plea with us to realize how profoundly we’re not seeing them – not letting them get as full and as strong as they can because we’re too busy trying to make them like they aren’t.  It’s not only disrespectful, this way we laugh them off and roll our eyes at them (“we always get tears when it comes to spelling!” and “boy, she always fights me on the boring parts of math!”).  It undermines our relationships with kids.  It lets them know in no uncertain terms that we are not available for communication; we are only interested in conveying our curriculum, whatever it may be and whether or not they take it in, whether or not they can use it. Any objections children may express that push us to reach beyond the scope of what we think they should know and do, we tend to dismiss as immature prattling.  We more or less laugh it off.

We don’t do it maliciously.  It was done to many of us, and so it comes naturally.  It feels natural and usual.  But that doesn’t mean it’s what we want for kids, and if it’s not, it’ll be well worth the effort it takes to leave it behind, to build  new, empowering, edifying traditions in its place.

Both, and; Milo

Two either-or traditions in education – that one must identify with one discipline over another, and must choose between learning for practical reasons and learning for its “own sake” – can really undermine progress toward the secure livelihood and fulfilled life most people want for their children.  

In Adam Gopnik’s recent New Yorker piece about the 50th anniversary of The Phantom Tollbooth, Gopnik writes that author Norton Juster’s story of young Milo’s journey was an argument “for the love of knowledge, against narrow specialization… for learning, against usefulness.” Continue reading