I like this little writeup on Nine Dangerous Things You Were Taught In School from Forbes. It pithily gets into the consequences of having a system that’s so standardized that is responsible for educating–a fundamentally intimate and nonstandard task, if you do it right.

I find myself in a place of tension on this topic. I believe in public education, and I believe we’re better off with it than without it for sure. On the other hand, I’m aware of the follies of a creaking factory-age education system and its numerous failures, including it’s  capacity for producing a kind of ignorance distinctive in those who have been systematically “educated.”

This comes down to some deep questions for me: is it possible to make space in the system of education for the real work of education? And if you do, how do you ensure that what needs to happen actually happens? And what actually is necessary?

Maybe this is a good time to mention that some people—like John Bennett, below—would go so far as to say that math, as it’s taught, isn’t particularly necessary. What do you think?

A nod of the head to Paul Salomon. Whose great blog, Lost in Recursion, is now in our blogroll. That’s where I found the Scale of the Universe link above, and also a host of thoughtful reflections on education and maddening problems like the one in the photo below:

Maybe the best way to drive home the ridiculousness of our relationship to the random is with comedy (thank you, Stephen Colbert, for the most mathematically sophisticated analysis I’ve seen on TV in a long time.)

Let’s review some facts about how random events work, which I’m trying to teach to 2nd-3rd graders at the moment, but which I’m pretty sure the majority of adults don’t know.

Fact 1: A random event is not affected by what comes before it.

This means that lottery numbers that have come up before are equally likely to come up again. The numbers that won last week’s lottery are as good a bet to play this week than any other set of numbers.* By the way, it also means that 1, 2, 3, 4, 5, 6 is as good a lottery ticket to pick (sorry, Colbert) as anything else*, so don’t disparage that choice.

Thus, ignore the advice I found on a randomly chosen lotto website: “Look at past winning numbers. If you really enjoy statistics and probability, look for patterns in these lottery tickets, and use that information for the best lottery numbers to pick.” Looking for predictive patterns in random noise is the ultimate fool’s errand.

*Of course, some numbers are better picks in that they are less likely to be picked by other people, so if you do win, you’re not too likely to have to share.

Fact 2: Nothing you do affects the outcome of a random event.

All your superstitions, knocking on wood, crossing your fingers, etc., don’t do anything. There’s a very strong human need to feel like we’re in control of our lives. Well, we’re in control of some things, but not as much as we’d like, and we have absolutely no control over which numbers come up on dice, cards, or spinning balls.

Now, there are ways to bias an event. I may be unable to predict whether it will be sunny or rainy on a given day next year (a random event, for all practical purposes), but my chances of a sunny day will be better in the summer. Random doesn’t necessarily mean even odds. Similarly, my students have experimented with spinning dice rather than rolling them to constrain the outcomes, and some claim that they can get their odds of a certain outcome up to 1/3 instead of 1/6.

Fact 3: Most specific events in the future are fundamentally impossible to predict.

Predicting the future in any precise way is impossible. Predicting weather in the future, predicting the stock market number precisely, predicting which horse will win a certain race, etc.–virtually impossible. Interestingly, if you change the question in the right way, it is possible to make much more accurate predictions. For example, I have no way of knowing whether a given coin toss will be heads or tails; however, if you flip 100,000 times, I’m confident roughly 50,000 of those flips (give or take) will be head. Similarly, I can’t predict the weather in the future, but I can probably get a reasonable read on the climate, and I have no idea what the stock market will do on a given day, but I may be able to give a pretty good assessment of where the economy as a whole is headed (though even though both of these questions are at the right scale, they’re still very difficult to answer).

The difficulty in predicting the future is a relatively new discovery, mathematically, but at this point, it’s been established in the field of chaos theory.

These are lessons I’d like the 2nd-3rd graders in my math circles to understand. I’d like them to let go of their superstitions, and try to understand how randomness really works. There is perhaps no lesson the next generation needs to learn more urgently. And here’s my prediction: those who understand probability and statistics will have huge advantages over those who don’t in the information economy, and those with a hard technical knowledge of both fields will always have job opportunities waiting for them.

I’m heading in a few minutes to meet with a group of kids that has been showing some real finesse for these ideas already. Lately we’ve been designing dice games to be tricky enough to appeal to others even though the games are biased in our favor. In other words, I’m teaching them to think like casinos. If I can succeed, maybe I can keep them from ever patronizing those places, too. Or buying a lottery ticket.

Because, face it, it’s just not a good bet.

For us, it was a very satisfying conclusion to a lot of preparation. Our plan from here is to make this an annual event, so keep your eyes open next year for the 2nd annual Julia Robinson Festival in Seattle!

That’s enough from us, though. Here’s what it looked like, and what attendees had to say:

“Our sons came home excited and jabbering away about the different activities. It was a great way for them to extend their math learning in a rare non-competitive way. We’d love to participate in future events.”

“Just a quick note to say — what an AWESOME job of preparation and coordination at this event. Even my kids were talking about it on the way home. And of course the content was interesting and fun. Thanks much!”

“Great festival! We’d love to see it take place every year. There are lots of math-minded families in this area which would appreciate it!”

“Could you make it longer next year? My son could have gone another hour easily Thrilled that you’re doing this!!!!”

“I had no idea what to expect and was so very impressed by this! I teach math in a constructivist way and loved how the mentors at each table let the kids puzzle and sit in disequilibrium, giving gentle hints when needed. The math challenges were awesome — and I loved how the table mentors were genuinely excited about math. I also enjoyed the “keynote speaker” i.e. the juggler. The morning passed very quickly!”

“Please make this an annual event!”

“Thank you for organizing this, it was an terrific event. We will come again next year and look forward to more Math4Love events.”

“From a fourth grade girl: ‘Awesome, would like to go again!’”

A huge thank you to MSRI, to the Evergreen School, to Greg Piper, Josh Zucker, and all our amazing volunteers, and to the Robinson Center, for helping financially, logistically, and doing all you did to make the festival a success!

For those in readers, the full link is here: http://mathforlove.com/best-of-the-blog/.

This is the Towers of Hanoi. The puzzle is almost intuitive: how can you move the tower from the left peg to the right without placing any larger disks on top of any smaller disks.

Wikipedia has this animated gif of a solution for four disks. A mathematical question that a natural followup: how many moves does it take to solve the puzzle with n disks?

A 4th grade student of mine took on a harder variation of this puzzle: how do you solve the puzzle–and count the number of moves–if you can only move disks one peg at a time. Notice that the gif above does not give a solution for this harder variation.

His solution was so excellent that I wanted to showcase it here. First of all, it’s a perfect example of how to use induction. Second, it’s personal–I love it when students use exclamation marks, because it shows that they’re really involved with the math as a story. Third, it’s concise. Fourth, it’s powerful. Fifth, pursuing this was his decision: he learned about the puzzle in Katherine’s How to Count Your Way Out of Trouble at the Robinson Center, and wanted to follow it to its full conclusion on his own. (If he hadn’t, I wouldn’t have suggested induction as a method of proof. ) This is exactly the experience of math I want my students to have.

I’ve often had a gut feeling that we actually invest life into math questions that grab us. Here’ s a question I like:

Question: How does a bishop behave on a torus chess board?

Many people, reading this question, might be turned off if the words are unfamiliar (bishops are chess pieces that move diagonally; a torus is what you get if you glue opposite sides of a rectangle together: like your computer screen in a game (for example, some Pac-Man versions) where you can move off the right side of the screen and come back in the left side, and ditto for the top and bottom). However, if the language isn’t in your way, or if you get past it (try http://www.youtube.com/watch?v=0H5_h-RB0T8 for example), there’s an interesting question here.

But even if you’re familiar with the language, the question may turn you off. It’s vague, after all: what does “behave” mean? “What do they want me to do here?” we can imagine the student crying.

Well, there’s no “they,” and I don’t want you to do anything. I’m just trying to share a question that is, for me, the launching point of a strange and exciting story.

So maybe I should just tell the story as it progresses for me.

Act 1: Stasis and disequilibrium.

A bishop sits on a chess board.

Casually, it looks off the right side of the board and sees that it is connected to the left side, and that the top is connected to the bottom. The bishop is the same (it moves diagonally), but there’s more to its world than it thought.

Not only that, the bishop is capable of different things in this different context. Suddenly the walls that prevented it from continuing its movement are gone. It progresses on the diagonal moving down and left to the bottom of the board and then through it. The old boundaries are gone! In fact, the diagonal is a kind of circle now, for when the bishop continues on its path, it wraps back to where it started. The world is different, the rules have changed.

How many squares can the bishop reach now? Which ones are they? What if it starts on a different square? Something in me wants to know.

If something in you does too, then stop reading this post, put your computer to sleep, and try to answer these questions.

Act 2: Escalating crises.

Once our character is out of its comfortable initial world, it is faced with a series of escalating crises. If we’re reading a novel, these come from the writer; here, they come from us. Doing math is partly a matter of writing the story as you go. This is one of the joys of the subject: you have more authorial control here than you can imagine. It’s also one of the difficulties: you can’t be a passive member of the audience. You make the bishop and the board happen. This is why you have to actually do the work if you want to experience the story in its fullest form.

If you have, maybe you know that the bishop can always reach the same number of squares wherever it starts (14 squares to be precise, counting the one it starts on). Then there are deeper questions: why is this true? The bishop is getting a sense of its power in these new settings.

There are the immediate questions I have, like–why 14 squares? How many would it hit on a smaller board, like a 6 by 6 board, or a 4 by 4 board? But these changes of scenery are precisely the new crises for our main character (the Bishop! Perhaps it’s because I played chess a lot when I was younger that I identify with chess pieces as characters. I like to chalk it up to my tremendous wellsprings of empathy). Suddenly, our question has widened: what exactly are the bishops powers when the board changes size?

And that’s just the beginning of the trouble! Once the board starts changing sizes, we have to consider the possibility that it changes shape too. What if the bishop lives on a rectangle? How many squares can it hit if it lives on a 4 by 6 rectangle for example?

At this point, our hero is starting to doubt facts about itself that it always took for granted. Nothing seems stable. Fortunately, we know some things about bishops for sure. For one thing, bishops, since they move diagonally, always stay on one color of the chess board: either they’re on the white squares or the black squares.

But, wait! Even the most cherished assumptions can be false! For on this board, the bishop moves from white squares to black squares. In fact, if you trace its diagonal, it’s actually attacking every square on the board!

Here, in the diagram to the left, you can see the beginning of the bishop’s move. On some board’s, the bishop is more powerful than anyone has ever dared imagine.

That may be a bit melodramatic, but I do feel a quickening of my pulse as the story continues. And the questions draw me in: when does the bishop hit every square? Is there any way to know how many squares it will hit?

Act 3: Climax and Resolution

The denouement comes with what some people call the aha moment, that instant of revelation where you see what’s really important when it comes to the bishop’s powers and all the other clutter and confusion falls away. As anyone who has felt it knows, it’s a magical moment. There are a lot of us suffering a lot of frustration because it’s so beautiful when that moment comes.

That’s why I’m not going to tell you the answer. Because with math, the most powerful endings are the ones you write yourself. And once you resolve the issues for yourself, the resolution is meaningful in the deepest kind of way.

Postscript: A New Stasis

Once you solve the problem, you experience some kind of mathematical version of catharsis, and what you’ve learned, both the facts and the arc of the story, settle into you and inform how you see and what you know. Bishops look different to you now, as do chessboards. But for the moment, things are settled.

But even this new stasis is deceptive. Didn’t we start with stasis? All we need is some new interruption, some idea to throw things into disarray again. Suddenly the world we know is ripe with possibility, ready to spin into disequilibrium as our curiosity nudges it.

Of course, maybe you don’t care for this problem. There are probably a lot of novels you won’t like either. But experiencing real stories, and real mathematical inquiries, is one of the most enriching thing you can do for yourself. Maybe this one doesn’t grab you, but one will, and when it does, you can be thankful that you are a human being, and human beings, for some miraculous reason, get to live their lives engaged by stories, by compelling questions, and by mathematics.

http://mathforlove.com/julia-robinson-festival/register/

Nickolai Pirak is a professional juggler from Seattle who will be giving a presentation at the festival on the relationship between mathematics and juggling. He will give an overview of how a numerical system of notating juggling moves called ‘siteswaps‘ has revolutionized juggling by providing a consistent language for jugglers to communicate tricks and patterns as well as create new ones.

Nickolai has been performing for over six years for a wide range of events and clients including Microsoft, The Bill & Melinda Gates Foundation, Seattle Art Museum and Sculpture Park, Woodland Park Zoo, Seattle Mayor’s Office, WA Lawyers for the Arts, The Northwest Emmy Awards, and The Moisture Festival.

Check out the videos to see Nickolai in action.

http://vimeo.com/37785552

And here’s the link: http://www.youtube.com/watch?v=VHWTS01uI9U