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.


Where’s *my* washing machine?

One of the commitments of modern education is equality.  We strive to provide the same high-quality education for every child. We do it by trying to give every child the same things.

My mom likes to tell the story of the year her parents bought washing machines for their two other grown daughters but not for her. My aunts had seven young children between them at the time.  My mom was single and moving around a lot.  On Christmas morning when the gifts were revealed, my mom said, hoping to get a laugh, “Where’s mine?”

I was thinking about how it would be perfectly reasonable for kids who are fascinated with and committed to things that don’t make it into the school curriculum to ask a similar question, though not in jest.  “How come the kids who love math get to spend 45 minutes of the day doing a thing they love, but the kids who love climbing trees don’t?”  “How come the kids who love to write get to spend 45 minutes of the day doing the thing they love while the kids who love fixing machines don’t?”

Of course our adult answer is that we’ve concluded that writing and math are the things that everyone’s going to need.  We’ve also decided that everyone has to get them at the same time.  But to kids, it’s just as uncomfortable and awkward to try to learn a thing it isn’t time for (even if it might be helpful later on) as it would have been for my mom to have a washing machine to contend with that year it was a tremendous help for her sisters.

I’m reminded, as I write, of the distinction between equality and and equitability.  Equal means the same; everyone gets the same.  It’s mathematical.  Equitable means “dealing fairly and equally with all concerned.” In those words, there’s maybe more room for each actual person to get what he or she actually needs.

Their own devices

It’s tough to use this phrase without getting derailed by the obvious pun or irony available given the various portable game consoles, MP3 players, and smartphones that populate many a modern child’s existence.  But I still often find it asking for my attention when I see young people at work on whatever is truly their own; when they’re left to what are actually their own devices –the mechanisms that operate in their minds and internal worlds, made visible in what they create and share with their speaking, drawing, singing, building, imagining, and other art and craft.

Yesterday I saw a series of drawings penned by a nine year-old I know.  One of what I would call this child’s own devices is a knack for telling terrifically dramatic and often ironic stories on paper, with spare line drawings and few words.  At first my eye was tempted to wince at the size and shape of her lettering.  And it would be easy to mistake what she’d drawn and written for an unsophisticated product for someone her age.  It would be easy to worry that she’s behind.

But the plots of these stories, the behavior of the characters, and the choice of words in the dialogue betray their author and illustrator’s wisdom and knowledge.  More than once as I was looking over the body of work I heard myself saying “I’ve never seen that done before.”

Kids’ own devices are often of this nature – a surprising and subtle confluence of the distinct neurological wiring they arrive with and the things they’ve seen and heard along the way that shape and inspire them.  When we’re distracted by how well they are or aren’t forming their letters or whether or not they can remember, quickly, the difference between 17 and nine, we can miss their best stuff.

Which is a shame, because it’s much easier to practice your letters once you find reason to do so, or devise a strategy for managing calculation, than it is to reclaim an authentically original and unique way of responding to the world after it’s been pushed aside or snuffed out all together.

Rules and tools

I was helping an 8 year-old with a math problem the other day.  It looked like this:

She pointed to Luke’s pencil and said “Well, it’s not this one because you aren’t supposed to start in the middle of the ruler.” She then proceeded to try to convince herself in various ways that one of the others (besides Luke’s and Nate’s) was right.

It’s possible that no one actually told her “you aren’t supposed to.”  Someone is more likely to have said “don’t,” by which that person meant “it’s easier math-wise to do it this way.”

But we don’t tend to say things that way.  We tend, in the interest of simplicity, we imagine, to say things like “don’t” and “you should” and so kids get into the habit of turning our words into rules.  Rules that in cases like this make tools (like rulers) more difficult to use.  Every time we tell kids how to do something rather than telling them it’s how we do it, we train them to hear things as rules.  It keeps them from investigating, and then they not only get stuff wrong on tests, they don’t learn how the stuff really works.

In the case of the ruler, if kids have the chance to notice that when you don’t start at zero you have to account for the extra space between zero and the beginning of whatever you’re measuring, they get to realize what the ruler’s actually doing. Rather than feeling, as many of them do, as though the ruler is  just another mystery invented by the adults, and their only chance of survival in the world of adult rules is to do as we say and hope they don’t forget one of the rules.  When they’ve had the chance to explore and really come to understand, they’re less likely to panic if one day a test question comes along, asking them to, like this one did, really notice and account for length. Not to mention that often in the course of real life activity (sewing, building, decorating), it’s actually easier to pinch a measuring tape at, say, two and 14 and hold the resulting 12 inches where you want them than it is to try to anchor the awkward zero end of the tape, and stop at the 12.

One way to help kids find their way to the kind of understanding that will allow them to use tools and knowledge effectively and comfortably is to acknowledge that when we show them how to do things, we’re really just showing them how we do things.

Room to grow

One summer I worked with a 10 year-old who’d been attending a small private school where her mother worked and was headed for public middle school in the fall.  Her parents were concerned that she wasn’t prepared.

This 10 year-old has an older sister, so she had an idea of what would be expected of her in middle school.  The thing she was the most worried about was missing the bus.  She knew herself to be distractible in the mornings, and did not want to be late for school.  So we got to work on a system (involving markers, clipboards, and clocks) that would prevent that from happening.

Each time her dad picked her up from my office, he’d  mention that he thought it would be good for us to work on multiplication soon.  Each time, his daughter would roll her eyes.

Finally one week she said firmly “Dad.  I’m not worried about the multiplication.”

“I don’t think you understand,” he pleaded. “Your classmates are going to have their multiplication memorized.  You don’t know yours by heart yet, so if you get asked in class, I’m afraid you’ll be embarrassed.”

“But I don’t care if I can’t do them fast, Dad. I know how to do them.”

I was reminded of that exchange this morning when I passed a family en route to school in my neighborhood.  The approximately 7 year-old child was wearing shorts and long sleeves.  The adults were wearing jackets and hats and the like (as was I; it was 39 degrees).  I heard myself thinking “He must be freezing!”

Just because I would have been. This is a habit we often fall into when it comes to advising young people – assuming that their experience of something is as ours would be. We don’t want them to be uncomfortable, so we tell them what we think they should do to avoid discomfort.  But there’s empathy – being sensitive to what someone else might experience – and then there’s projection.

When we’re careful not to mix the two (or at least own up to and fix it when we do), we make it possible for kids to have their own experiences, and even to get stronger than we ever got.

My young middle school-bound friend was confident in a way that her dad knew he wouldn’t have been under matching circumstances.  She was socially comfortable enough to know that anyone who’d judge or tease her for taking a few seconds to find her way through 6 times 8 wasn’t someone she was going to want as her friend.  And that she would find the ones she did want.  Her dad had played a part in raising daughters with this kind of confidence, and once he realized that she was OK with her slow multiplication, he could let her boldly tread where he wouldn’t have dared.  He’d called her attention to the area of possible discomfort, so he knew she’d been fairly warned. From there she could choose how to manage it. As it turned out, she had a confidence and ease he hadn’t had at the age of 10.  Which of course was exactly what he always wanted.

Will it get in the way?

I have an acquaintance who’s a published author but never quite got the hang of spelling.  And a friend who’s a research scientist with a PhD who has never been able to add or subtract very well.

Spelling and quick mental computation can be helpful, without a doubt.  But even for their highest-order relatives (like professional writing and scientific research), proficiency in some things we call basic are not necessarily necessary.  What we often forget to ask ourselves, when kids are struggling, is whether or not the absence of a particular proficiency is going to get in the way. Is our attention in the places where we actually want it?  Are we sometimes (or often) distracting ourselves and kids from pastimes that will prepare them for what they’re best suited to?  I’ve quoted Stuart Brown before, regarding a discovery of the Cal Tech Jet Propulsion Lab:

“Cal Tech’s Jet Propulsion Laboratory (JPL) has been the United States’ premier aerospace research facility for more than seven decades… [In] the late nineties, the lab’s management was saying, ‘JPL, we have a problem.’ As the lab neared the new century, the group of engineers and scientists who had come on board in the 1960s, those who put men on the moon and built robotic probes to explore the solar system, were retiring in large numbers. And JPL was having a hard time replacing them. Even though JPL hired the top graduates from top engineering schools like MIT, Stanford, and even Cal Tech itself, the new hires were often missing something. They were not very good at certain types of problem solving that are critical to the job.  The experienced managers found that the newly minted engineers might excel at grappling with theoretical, mathematical problems at the frontiers of engineering, but they didn’t do well with the practical difficulties of taking a complex project from theory to practice…. They found that in their youth, their older, problem-solving employees had taken apart clocks to see how they worked, or made soapbox derby racers, or built hi-fi stereos, or fixed appliances. The young engineering school graduates who had also done these things, who had played with their hands, were adept at the kinds of problem solving that management sought. Those who hadn’t, generally were not. From that point on JPL made questions about applicants youthful projects and play a standard part of job interviews.”

It’s not that offering a child the chance to learn to spell and calculate has to mean that that child won’t have the chance to play, or problem-solve, or the myriad other things there are to do in the course of a day.

But imagine a child who is quick and accurate with mental computation, because that’s where we’ve put our emphasis, but the child can think of only one way to solve any given problem.  She’ll get all our accolades as a young student, all the support and encouragement she’ll need to be successful throughout her young academic career.  But as an engineer?  Or as a doctor?  She’ll be missing a key ingredient.  Or a child who spells flawlessly, never misses an editing mark on his daily proofreading exercises, can organize a paragraph with precision, but doesn’t much like stories.  He’s encouraged to major in English, goes on to study creative writing or journalism.  But he’s missing a key ingredient.

What do they actually need, and will the things we’ve always held as paramount really hold kids back if they’re not mastered?  Answering the question requires a bit of imagination and a lot of pragmatic perspective.  What are these things really useful for?  What do they actually bear on?  And what are the actual building blocks for success and security?  My author and scientist friends would tell you it’s not spelling or computation.  They’re living proof of that.

This could feel like good news or bad news.  I think it’s good news, if it means that we can let go of the traditional mandates that wreak such havoc on the confidence of young people and waste so much of their time, in the often false name of preparation.  It might feel like bad news in the sense that it means we can’t just keep sending them to school, trudging them through standardized progressions of academic material. At least not if we really want them to be prepared for (and inspired to go after) the kinds of livelihoods that are actually out there. It’ll take more in the way of imagination, and actual information gathering.  We’ll have to look at what it really takes to make it in the world today, what actual jobs and professions require of their employees and participants.  And then we’ll have to invent what the pathways to those true requirements might look like.  Lots more effort, but if it means we end up with young adults who are truly prepared, not just academically apt, such that they can get busy solving problems, finding purpose, supporting themselves, it’ll have been worth it, and then some.

Refusing to fake it

Yesterday a 10 year-old said this to me about her experience with math: “It’s like I’ve been doing it, but I haven’t been learning it. People keep saying ‘well if you’re doing it you must be learning it,’ but I don’t think I am.”

What she meant by doing it was that she was performing the tasks that constitute math in her classroom and what’s assigned for homework, but she didn’t understand much of it.  And she thought learning should be about something more than getting through an assignment.

In many cases, the old “learn by doing!” axiom is a helpful way to think about things; it tends to be much more effective to give someone the chance to actually engage in the activity of something than to just tell them about it.  But when we ask kids, and other people, to go through the motions of something without context or conceptual grounding, we drain the learning out of it.  We make it about performance and not about growth.

Every once in a while someone notices, and demands more.