Just kidding

Andrea* looks down at the algebra problem she’s working through and notices that she’s assigned a value of five, rather than six, to three twos. As she erases the five and replaces it with a six, she says “Just kidding.” We both smile. Then she continues with the rest of the problem.

This is a simple but brilliant little practice of hers.  Math can be so charged, and the prospect of making a mistake in math inspires fear and trepidation throughout the land.  If a young person can relate to miscalculation as an opportunity to pretend they’ve made a little joke, they’ve got at least one way to keep perspective.

So many of the kids I work with have learned to tense up and start defending themselves when they can’t remember something or when they mix things up.  Their eyes dart up to see how I’ll react, and before I even have a chance to, they start spinning their talking wheels – “Oh, I thought we were supposed to do plus, not times… My teacher said… When we did it in class… This is so confusing…” Or they just give up all together and tell me they can’t do it.  Usually over something as small as five, instead of six, for two times three.  These kids have received the message that if you don’t get every bit of it right every time, especially the single-digit stuff, then you might as well hang up your math cleats and plan on a route that doesn’t include any numbers.  They expect to be judged on their ability to achieve computational perfection.

Andrea figured out, in time, that it’s possible to miscalculate, even often, and still excel as a math student.  And that if she keeps her sense of humor about her, she can keep her head in the game.

I’ve started telling the younger kids I know, especially those who get skittish when they mix up six and five (or write a seven open to the right instead of left), about Andrea’s just kiddings.  I’ll say something like “One of the teenagers I know, when she makes a little mistake like that, always says ‘Just kidding.’ She’s not saying that to really pretend she meant to do it, she’s saying it because it’s funny to pretend she meant to do it.  I think she does it to remind herself that making a little mistake is no big deal and if she makes a little joke about it, the mistake doesn’t distract her from the real thinking she’s trying to do.”

A couple of them have tried it, and with noticeable results.  It interrupts the habit of panic and doubt, creates a space for relaxation and ease.  And there’s nothing like a little calm to free up the mind for math.

*Not her actual name.

Forced math love

The heading of the article reads “Learning to Love Math.”  My pulse quickens for a moment.  I like the sound of this. From personal experience, I know that it is possible to learn to love math. When I was 8, and 9, and 10 years old I’d have told you I hated it.  Then I got the hang of it (or maybe something changed about the way it was offered, or even what was offered as it), and I started to like it.  Later still, it became something I would think about voluntarily, something to do for fun.  And now sometimes I get to share my love of it with other people, and then it’s fun again, and more.

So when I came across this article about learning to love it, I read on with excitement.  But then I got to this part, explaining a professor’s mission in rethinking math education: “We need to teach kids to love math, not just to get through math.”

While I agree entirely that it’s better for everyone if we come from an intention of inspiring love, rather than settling for the survival of “getting through,” the use of the word “need” left me a bit disappointed.

Every time we decide that we have to teach someone to love something (reading is another place we demand this of ourselves), we make the work of sharing knowledge and skill more difficult for ourselves and the task of receiving it more difficult for those with whom we intend to share it.  To show kids how something like math can be loveable is indeed more effective than just shoving boring-ified things down their throats.  Much more effective.

But to demand of ourselves that we get every person to love one thing is to doom ourselves to failure.  It’s just not possible. People are not like that. We’re not all going to love the same things. And further, humans (children included) are more available for learning when we don’t feel as though we have to take on someone else’s experience of the content, or someone else’s expectation of how it should seem, feel, be appreciated or used.

And we don’t have to love things in order to use them for what we’ll need them for. With the same commitment (to revealing the beauty of math and other potentially useful and loveable things), we could say things like “If we expect kids to be able to understand and use math, we should stop turning it into something that feels disconnected and arbitrary.” I know that’s probably what the quoted professor mostly meant.

But the words matter, and our longstanding habit of using the insistent language of “have-to” when we talk about young people and education is not without cost.  It’s possible to use language about math and other realms that won’t force us to face off with the diversity of human preference. We can choose words that make room for us to draw the potential appeal forth from the numbers (or the books or the music or the carpentry), words that will let us look for ways to make things feel more humane,  attractive, and accessible without insisting that those things occur the same way for everyone.

Math resources, one new and one remembered…

I’ve added two resources to the math section of my recommended book list on amazon.

The first is two volumes’ worth of compiled Math Olympiad problems.  I’d forgotten about these, but I like them because there’s usually more than one way to solve each problem and the solver doesn’t know what kind of problem to expect at the outset.  You can see sample problems here; they come in “elementary” and “middle school” levels. (It may look as though I’ve botched the links, but I haven’t.  If you order any, be careful to examine the covers before choosing.  Volume 1 is called “Contest Problems for Elementary and Middle School” but it only contains elementary level problems.  Volume 2 is called “Contest Problems” but has some of each level. I think it’s just because when the first volume came out there was no second volume, and the title was describing the program, not the contents of the book.)

The other is a series of beautifully designed books from Thunder Bay Press called Doodle Yourself Smart. There’s a math volume, a geometry volume, and a physics volume.  Each book has 100 or so puzzles, one per page with lots of room for figuring (and doodling, ostensibly).  They’re refreshingly well-designed, for math books. The pages are actually pleasant to look at.  I’ve only interacted directly with the math volume (the puzzle I did was about finding pairs of primes that added up to various target numbers), but I’m looking forward to the other two.  I’d recommend these especially for math-likers with aesthetic sensibilities.

Enjoy…

Penmanship

I know several kids who write very, very slowly. I know others who like to decorate their letters as they write, many who form their letters starting at the bottom rather than the top, and lots who despise the task of holding a writing utensil at all, complaining of tired and weak muscles.

I watched one of these slow writers doing some math the other day.  The speed of her math performance has been a point of concern and discussion in school lately. It occurred to me as I was watching that part of the reason she takes a long time getting through math problems is that she wants the numbers to look nice.  For her, writing numbers (and anything else) is an opportunity to make art.

Artistry is often at work with the letter-decorators I mentioned too, though I’ve also seen letter-decorating used primarily to combat boredom.  Here are two other interesting coincidings: those writers who work from the bottom of the letter also tend to be the ones who would rather be designing and building things than sitting bent over a piece of paper, and the messiest and most apparently tormented or resistant are often the ones to whom the words are the most important.  The writers.

I’ve been observing young writers for a long time, and I was also one myself once.  The year I was eight was significant for me. I spoke in front of a large group of people for the first time, among other things. But the thing that got the most attention that year was my handwriting.  It wasn’t very good. I was in too much of a hurry, the adults told me.  I could do better.

Fortunately, that flurry of concern over my sub-par penmanship didn’t leave much of a mark on me, as far as I can tell.  I know that the parents and teachers who harped on my letter formation back then had my interests at heart and in mind.  I’m pretty sure that if they had realized I was just trying to keep up with my thoughts, they’d have handled it differently. The teachers I know now aren’t as hard on kids about handwriting as the ones I had when I was young, but we still tend to miss the opportunity to learn from what goes on with kids when they sit down to write – not just how the letters look but how kids are about it and what communication there may be for us to receive in the course of watching.

We miss this opportunity for noble reasons; we believe we know how to tell when writing’s going well and when it’s not.  The sight of neat legible letters soothes us, makes us feel as though things will be OK for the child forming those letters.  But too much haste, too little haste, unusual pathways, and general resistance worry us.  The task of writing feels important, so we get rigid and frightened about it and push for the results we know to push for.

But being rigid and frightened makes it hard to see what more there is to see, and it tends to undermine access to the very proficiency we’re after.

Here’s the thing.  The word is penmanship.  As with craftsmanship or sportsmanship, there’s grace and individuality suggested by and allowed for in the word.  Penmanship has come to refer only to how tidily we write, but it didn’t start there and we don’t have to settle for that.  We can ask ourselves more interesting questions about the emerging penmanship(s) of those newest to the tool – the way each one wields his or her pen.  What is there to see in a child’s resistance to writing?  What might it lead to?  Why would a person spend as much time drawing spiraling tails on every letter as choosing the words the letters make up?  Why is the messy writer in such a hurry?

If we ask questions like these, we’ll get insights into the behaviors themselves and also, most likely, surprising causes for further curiosity and even celebration.  And we’ll make lots more room for young people to come to own the work of writing, and to call on it to serve and support them in whatever paths and pursuits they choose.

Fun with circles and teeth

A friend handed me one of these the other day and said “Like Spirographs, remember?”

I didn’t remember, but if I ever used one I’m sure I loved it (and apparently it’s possible to find similar products now but Hasbro doesn’t make the original anymore; if I’m wrong please let me know!). This thing is great fun if you like shapes or patterns or color or the unexpected.

And it’s one of those things that the mathematicians like to play with and calculate about that blurs the boundaries between math and art. I’m pretty sure this is the kind of thing Paul Lockhart is talking about all the time; the math that tickles, and wrinkles the brow with amazement.

I mentioned a talk by Conrad Wolfram a while ago, which I remembered in the midst of playing with the hypotrochoid set because I went looking to find out how the thing works and found this animation on his MathWorld of a point rolling around inside a circle in a fixed way which is what goes on with hypotrochoids.  No numbers required in the marveling at it…

Paperweight

I was once asked to tutor a ten year-old who didn’t want any help.  (This has happened lots of times; I say “once” because this story is about one particular child.)  He was a relatively good sport about it, because he’s a relatively compliant kid.  He was not about to refuse to meet with me, and he was not about to be rude to me.  But it’s hard for anyone in a situation like his to go without an outlet for resistance.  So this is what he’d do.  When it came time to write anything down on a homework assignment he’d write with one hand but not steady the paper with the other.  The result was nearly illegible numbers and symbols.  Just generally a big mess.

I’ve seen enough children doing this to discern with some accuracy when it’s the result of a lack of understanding of the physics involved in the act of writing (which it really sometimes is) and when it’s a communication.  This was a communication.

I could have told him to hold the paper still, and he probably would have obliged (being relatively compliant). But then he would have found some other way to let me know he wasn’t happy with the circumstances.  Instead I asked him if sometimes he holds the paper still when he’s writing on it.  He didn’t respond right away.  “What d’you mean?” he said (with what sounded to me like caution).  I said, “I mean, I’m pretty sure you know that when you’re writing, and you hang on to the paper with your other hand or steady it with your wrist, what you write will be easier to read.  So it seems like maybe you don’t feel like it right now, or something.”

He didn’t say much then, and we moved on. But the next time it happened, when he realized he was doing it, he looked up at me and I raised a dramatic eyebrow.  He covered his eyes for a moment, scrunching up his face, and laughed. It became a bit of a running joke between us.

My approach didn’t change the fact that this child didn’t really want to be there with me working on math, but it set a tone that allowed us to talk about it, person-to-person.  And that meant we could also talk about the various challenges and resistances that led his teachers and parents to send him to me in the first place.

This is one of those things that can seem simple but not actually be easy.  It’s often not easy to figure out how to acknowledge out loud that a child’s will is involved in a behavior, with curiosity about the behavior and without immediately attributing the expression of that will to laziness or obstinacy. But it’s possible.  And it’s worth it.  When we find ways to access and express genuine curiosity about why kids are doing what they’re doing, we make room for a human connection that transcends the common adult/child dynamic – the one in which an adult gives a directive of one kind or another and the child is limited to a choice between compliance and defiance. Breaking the cycle of that dynamic tends to allow for much more productive and peaceful conversations.

And it’s also just plain more fun and less exhausting for everyone.

The guys who landed Curiosity

Since the rover Curiosity landed on Mars in August, two of the engineers involved in the project have been getting lots of attention in the press.

Adam Steltzner, who was in charge of the actual landing, has been telling the story of his unorthodox path to accomplishment in the aerospace field.  According to Steltzner, he wasn’t much of a student in high school, so he stopped going.  A few years later he was driving home at night and got to wondering why the stars he could see were in different positions in the sky from how they’d appeared earlier in the evening.  Soon after, he attempted to enroll in an astronomy class at the local community college but found that the class had a physics prerequisite. Steltzner had struggled with basic high school math, so one might imagine that the prerequisite would have ended his quest for an astronomy education.  In fact, it was the beginning of his ascent to tremendous success as a scholar in and as a practical contributor to aerospace engineering.  His alma mater captured the phenomenon this way: “Steltzner quickly experienced the epiphany that has transformed many lives before his: What people resist doing by rote and requirement, they’ll cheerfully embrace through passion and curiosity.”

Steltzner’s colleague Bobak Ferdowsi, the flight director on the Curiosity mission, traveled a less circuitous route to his occupation, but his success as an engineer also seems to have begun as Steltzner’s did, with a fascination with science and space.  He told it this way to WIRED: “I always loved science fiction, I used to love to draw spaceships. Another thing that helped me as a kid was that I played with Legos constantly. I’m sure a lot of kids do, but for me it was not only being creative but being able to build the thing that you’ve imagined. It’s hands-on engineering. We actually use Legos here at work sometimes – more in the early part of the mission – when we’re trying to make a quick 3-D model of something. Legos are one of the reasons I ended up where I am.”

These two would not likely be as successful as they are without the rigorous academic training they’ve received.  But for both of them the prelude to that academic training was critical.  Academics alone was not enough to engage Steltzner.  He needed context and inspiration before it felt worth it to apply himself sufficiently to the study of the field in which he’d go on not only to succeed but to innovate. And from the sound of it, Ferdowsi didn’t struggle as Steltzner did, but he makes it clear that the Legos and the spaceships of his youth were instrumental.  His context and inspiration just came earlier than Steltzner’s did.

The role of the sky and the Legos in Steltzner and Ferdowsi’s accomplishments might seem like grounds for a mandate of stargazing or building.

But these guys aren’t saying “Thank goodness someone made me learn this stuff.” They’re saying “This is what fascinated me. Then I went after it.”

When we start spending our energy and other resources supporting young people in finding their own fascinations – their own versions of the stars or the Legos – rather than only on convincing them how important it is to read and write and calculate early, we’ll end up not only with lots more accomplished and knowledgeable people in every field, but also with more proficient readers, writers, and mathematicians. For so many of us, maybe even most of us, the context makes all the difference.