Mike South on The Meaning of Zero

I'm on a google group called NaturalMath, hosted by Maria Droujkova. We were discussing how it can take some deeper thinking to understand zero.

One person wrote:

Maybe because we use [zero] as a symbol for the abstract concept of nothing, which none of us have experience with, it prevents other considerations...

Mike South's reply struck me, and I asked him if I could report it here. So you guest blogger for today is Mike South...

Discovering Zero

I don't know--we don't have experience with circles, either--not perfect ones--but I think we can imagine them pretty well.

I can ask myself "how man people are in the room besides me?" and want to have something to call that when the last one leaves. But I think that particular idea is post hoc. I think people discover zero when they realize they need it for something. Similar to negative numbers--not something people (generally) sit around and think up, but they come up naturally.

I would guess that most of the times people discover a need for zero (not that we let them do that any more, preferring to push it on them when we think it's time) it comes in a positional number system like the dude on the wikipedia page who just kind of "needed something to keep the other numbers lined up". Eventually they figure out that there is a perfectly logical way of thinking about it as a "real" entity and not just a made-up mental crutch to get you to the "real" answer.

Time is a possible example where you could start thinking of zero as something real. You can represent two days from now as 2, 1 day from now as 1, 1 day ago as -1, 2 days ago as -2, and then say, well, what about today? When it's "no days ago" or "no days from now", what do we write for that? You can say "well, that's just today", and I think people probably normally start there. But eventually it creeps into your mind that maybe that ought to be a number in its own right.

European-style floor numbering is another good example, I think. In (at least some) countries in Europe, they number the floors based on how far up you are from street level. So what we would call the second floor, they would call something like "one floor up". And if you number one floor up on the elevator buttons as "1" (and they do), and 2 floors up as 2, well, one floor down (say you have a basement parking garage) is -1, 2 floors down is -2, what do you label the button for the ground floor? You could put a "G" or "M" for main floor, but that should eventually get people thinking--why no number for the threshold between up and down? You can probably go for a long time saying "it's not up or down. It's just ground level, and that's why we call it that--you don't need a number because there is no distance." And in some sense that's simply correct.

But once you let that little fellow creep in to your mathematical vocabulary, it ends up being indispensable and making things way, way easier to deal with. And that is probably ultimately what ends up being thought of as "real"--if it's useful in many situations, if we are mentally mapping either abstract problems or physical realities onto it, we end up wanting it into existence.

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Mike South has written a few more things you might like, available at his Fulcrum site.

Links: Math Anxiety news, Games, and Arches

• Most elementary teachers, like most adults in the U.S., are uncomfortable with math. That affects all the kids, but here's a new ABC report that it may affect the girls more.
• Logic mazes, yum. (Thanks for the pointer, Owen.)
• Cool video about how arches got their shape.

My son's board game :^)

My son, we'll call him R, goes to Wildcat Community FreeSchool, where the kids get to play most of the day, and learn lots of good stuff that way. Wildcat has lots of voluntary, free form classes, like art, Spanish, music, and other impromptu things, and there are also semi-mandatory classes in language arts and math. (If the parent says they're taking care of these subjects at home, the child doesn't have to go.)

R has been uncomfortable in crowds for the past year or two, and has hated 'having to' go to class. So he didn't. We do plenty of reading at home, and I'm all about math. But the classes are pretty fun, and he finally started going last week. (I don't know that it will last.) I'm thrilled. I trust that he'll learn just fine without it, but he's very social, and I think he'll be happier going to class.

On Thursday, the teacher (go Sarah!) had the kids rolling dice and marking their results on a chart with maybe 8 boxes above each of the numbers 1 to 6. Doesn't sound particularly exciting, does it? But that's because you're not 7 years old. The kids were watching the numbers race to the top. They were fascinated.

Yesterday evening R was messing around with the dice, and suddenly said, "I can do the game Sarah showed us!" And he made his own chart and raced the numbers. (Meanwhile I'm sitting near him, reading all the blogs I follow.) When he got tired ot that, he made his own board game:

We played it together today, and I loved it! Yes, it's utterly simple. But it was amusing what happened sometimes. The image doesn't show up very well, so I'm going to bore some of you with the details: Space 0 says 'Start', 1 is back 1 (indicated by arrow), 2 is back 2, 3 is forward 3, 4 is lose a turn (frowny face), 5 is back 2, 6 and 7 have no special actions, 8 is back 6, 9 to 12 - no actions, 13 is back 7, and spot 14 (not visible) is the goal. Landing on spot 8 takes you back 6, and then back 2 more to the start. Landing on spot 5 takes you back 2 then forward 3 to spot 6. It was a topsy turvy game.

This is what I want for my son - enough exposure to math that he has a glimpse of its power, and enough freedom to play with math however he wants. Yeay!

Richmond Math Salon and Base Three

I've been doing this for a year and a half now, and it's finally beginning to gel. There are some anchor families who come most of the time. There are new families coming. I'm starting to be able to describe how important it feels to me to work with parents and kids together.

Most of the folks coming to the math salon are homeschoolers. It's even more important for them than for others that the parents enjoy math. But, really, every kid, whatever sort of schooling they experience, will have a richer experience of math if their parents enjoy it.

But most parents don't. Most people over 6 or 7 or 8 don't. (You know, after they've had enough arithmetic lessons to think it's either boring or senseless.)

So if they can bring their kids to a math salon or a math festival or... and work on something together, then the parents can relearn how to have fun with math.

Today I had a sign up on my front door:

Enter at your own risk!
This doorway takes you
Through a spacetime warp,
Onto the planet of Triplay.
Are you ready?!

When people come in, there are games like Blink!, Set, Rush Hour, and Quarto to help them settle in. And Lori, the mom in one of the families that has come often, was there to assist me, since I was expecting a larger than usual crowd, with younger kids. (Although I've had months where tons of people are going to come, and then half of them can't make it.) Lori's daughter Audrey helped me too, sharpening pencils and playing a game with her mom, so mom would be ready to explain it to other parents.

We did this for close on an hour, I'd guess. Then I gathered all 20 of us in my small living room (except the girl who was sleeping in a bedroom). I started us off by talking about the people on Triplay having only 3 fingers on a hand - we all held up our hands, either with two fingers folded down, or fingers grouped in the Star Trek (live long and prosper) way.

We began discussing how the people of Triplay (Triplatians?) count: one, two, hand, hand and one, hand and two, two hands. We were all holding up our hands while we counted and talked about it. When we got to three hands, we needed a new word. We voted on whether our word would be handhand or handred. We voted for handhand - the cuter one. But then we got to four hands, which is handhand and hand. Too confusing! So I talked about how our language evolved over time to handred.

We kept counting, decided that 3 handreds was a handsend. (We didn't write it down at first. When we were writing these down later, it was interesting to think about how they might be spelled. It felt like a linguistics exploration too.)

The kids started drawing people from this planet, and some of the adults drew too. After a while of that, we began to design the Triplatian monetary system, and came up with names for the coins. I had a bunch of plastic chips available to use as coins, but I'm not sure how much use they got. We came up with our own names for the 3 cent piece (trickel), the 9 cent piece (handrime) and the 27 cent piece (hansellar). I'm not sure what the 81 cent piece was named.

A few of the adults began working on how numbers are written on Triplay - 1, 2, 10, 11, 12, 20, 21, 22, 100, and 101 are the first ten numbers, if you use our place value system and their base (three). They did that with no suggestion from me, so I was all the more excited to see the work they were doing. Two of the people working on that had mentioned having traumatic childhood experiences at home with math, so you know that diving in like that took some courage. (Isn't it amazing sometimes how our efforts to parent well push us to grow in ways we never expected?)

As usual, the adult conversation eventually meandered from our activity over to what coop classes they've found, and questions about books. I pulled out at least a dozen books, and we talked about how they do things in China (read Liping Ma's book, Knowing and Teaching Elementary Mathematics: Teachers' Understanding of Fundamental Mathematics in China and the United States, for the good stuff, but we also talked about how pressured kids are there, as in many countries).

This was the best math salon I've done yet*. Other sessions there've lots of people, and things seemed to go well, but this time it felt like we really had a great balance of individual and group exploration, and I was tickled that I've gotten better at working with younger kids (drawing pictures, yeah!). Four people came later, so there were 24 people in all - the most that have ever come to the math salon.



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*I've had lots of math salon sessions flop, mostly due to low attendance, but can't find the energy or courage after those to write. I'm so impressed by the math teachers who blog about what they're struggling with. I will try harder next fall, when I'm back to teaching, to tell you all about some of my bad days.

Sleep and Learning

I sleep about 9 hours on a good night. (Sometimes I wake up in the night, start thinking too much, and can't fall back to sleep.) My 7-year-old son sleeps 11 hours a night. I've always hated waking to an alarm, and haven't had to do that for years now. We go to bed between 8 and 9 pm, and if my son sleeps late, I let him. (His school is very unusual, and it's ok for him to arrive late.)

I'm happier - and I can teach better - when I've had enough sleep, so it's been a priority of mine for many years now. Recently, I've been hearing about how important sleep is for our health and our brains. The article I read just now (Snooze or Lose, at NY Magazine) gives even stronger evidence than I've seen before. This paragraph got me started writing this post:

In Edina, Minnesota, an affluent suburb of Minneapolis, the high school start time was changed from 7:25 a.m. to 8:30. The results were startling. In the year preceding the time change, math and verbal SAT scores for the top 10 percent of Edina’s students averaged 1288. A year later, the top 10 percent averaged 1500, an increase that couldn’t be attributed to any other variable.

A combined score of 1500 is in the top 1% of all test takers. 1288 is in the top 15 to 20%. This is a huge jump. The article gives detail about how sleep helps us learn too:

Dr. Matthew Walker of UC Berkeley explains that during sleep, the brain shifts what it learned that day to more efficient storage regions of the brain. Each stage of sleep plays a unique role in capturing memories. For example, studying a foreign language requires learning vocabulary, auditory memory of new sounds, and motor skills to correctly enunciate new words. The vocabulary is synthesized by the hippocampus early in the night during “slow-wave sleep,” a deep slumber without dreams. The motor skills of enunciation are processed during Stage 2 non-rem sleep, and the auditory memories are encoded across all stages. Memories that are emotionally laden get processed during R.E.M. sleep.

At the end is a companion article about how to get more sleep. The bit of advice I found most helpful is to limit exposure to tv or computer screens in the last hour or two before bed. I guess if I want to sleep more soundly, I'd better stop surfing a bit earlier.

Sweet dreams!


(Kudos to Rebecca Zook, who posted about this in her blog, Triangle Suitcase.)

Challenge: Write a Kids' Poem about Math

Back in September, Sean Nash wrote a post, over at his nashworld blog, about reading a book of Mother Goose to his 2-year-old daughter and not reading this one:
She's recognizing words, maybe he's lucky she let him skip it. I wonder what the rule of three was back then.

Reading the comments, including one from a dad who's encouraging his daughter to write science poems, got me thinking about what more we could do. So...

Here's the challenge: Write a little kids’ poem that’s as catchy as that nasty bit quoted above, and that tells of the beauty of math, or, that mentions math and challenge, both in a positive way.

If you start working on this, please let me know, even if you don't come up with something you like. I know I need to let this thought percolate for a while before I'll be able to come up with anything.

Joint Mathematics Meetings in SF Next Week

I'll be attending. If you'll be there and would like to meet up, please email me. I'm suevanhattum on hotmail.

I'm planning to buy a few books I've heard good things about:
Euler's Gem, by Dave Richeson, who blogs at Division by Zero,
The Calculus of Friendship, by Steven Strogatz (read Sam Shah's review),
and some of James Tanton's books.