Math is an art??

Okay, I've always held a vague idea that there is a way to explore mathematics that is really interesting and fun; every once in a while I've read a quote or a snippet from an article that hinted of the magic to be found in exploring mathematical relationships, for example, and it's been on my to-do list for a couple of years to look into it.

Our friend Jill posted a link to this article (click on the link towards the bottom of the page "Lockhart's Lament" for the article) on our homeschooling group last week and I'm absolutely fascinated by what this mathematician/educator has to say. I'm realizing that I've been duped into believing that math is about numbers and calculations, not about spatial relationships, leaps of insight, playfulness and art. I'm not finished with it yet (it's 25 pages long!) but it's challenging all my assumptions about what math is and how it should be experienced! In particular, his explanation on pages 3-5 of how math is an art form rather than a practical application was quite paradigm-shifting for me....

Coincidentally, Steve came across another article in an issue of the New Yorker last week that explores the fact that we now know humans have an innate feel for approximate math and it can get quite complex, but that how we talk about and teach math is very non-innate and hampers our understanding. English is among the languages that does a bad job of communicating math (cantonese, I think is the best, for its simplicity of naming numerals) and while addition and subtraction are intuitive enough, multiplication and division are outside of our hardwiring.

One of the fascinating suggestions both mathematicians raise is this, (the quote is from the New Yorker article):

Our inbuilt ineptness when it comes to more complex mathematical processes has led Dehaene to question why we insist on drilling procedures like long division into our children at all. There is, after all, an alternative: the electronic calculator.
"Give a calculator to a five-year-old, and you will teach him how to make friends with numbers instead of despising them," he has written. By removing the need to spend hundreds of hours memorizing boring procedures, he says, calculators can free children to concentrate on the meaning of these procedures, which is neglected under the educational status
quo."

Lockhart, the author of the 25-page lament, writes in a dialog form that is a nod to Galileo's writings on science and mathematics in which three distinct voices debate (bless you, participants in the world of Wikipedia! I would never have figured out why he was using this construction otherwise!) :

It would be bad enough if the culture were merely ignorant of mathematics, but what is far worse is that people actually think they do know what math is about— and are apparently under the gross misconception that mathematics is somehow useful to society! This is already a huge difference between mathematics and the other arts. Mathematics is viewed by the culture as
some sort of tool for science and technology. Everyone knows that poetry and music are for pure enjoyment and for uplifting and ennobling the human spirit (hence their virtual elimination from the public school curriculum) but no, math is important.

SIMPLICIO: Are you really trying to claim that mathematics offers no useful or practical applications to society?

SALVIATI: Of course not. I’m merely suggesting that just because something happens to have practical consequences, doesn’t mean that’s what it is about. Music can lead armies into battle, but that’s not why people write symphonies. Michelangelo decorated a ceiling, but I’m sure he had loftier things on his mind.

SIMPLICIO: But don’t we need people to learn those useful consequences of math? Don’t we need accountants and carpenters and such?

SALVIATI: How many people actually use any of this “practical math” they supposedly learn in school? Do you think carpenters are out there using trigonometry? How many adults remember how to divide fractions, or solve a quadratic equation? Obviously the current practical training program isn’t working, and for good reason: it is excruciatingly boring, and nobody ever uses it anyway. So why do people think it’s so important? I don’t see how it’s doing society any good to have its members walking around with vague memories of algebraic formulas and geometric diagrams, and clear memories of hating them. It might do some good, though, to show them something beautiful and give them an opportunity to enjoy being creative, flexible, open-minded thinkers— the kind of thing a real mathematical education might provide.

SIMPLICIO: But people need to be able to balance their checkbooks, don’t they?

SALVIATI: I’m sure most people use a calculator for everyday arithmetic. And why not? It’s certainly easier and more reliable. But my point is not just that the current system is so terribly bad, it’s that what it’s missing is so wonderfully good! Mathematics should be taught as art for art’s sake. These mundane “useful” aspects would follow naturally as a trivial by-product. Beethoven could easily write an advertising jingle, but his motivation for learning music was to create something
beautiful.

SIMPLICIO: But not everyone is cut out to be an artist. What about the kids who aren’t “math people?” How would they fit into your scheme?

SALVIATI: If everyone were exposed to mathematics in its natural state, with all the challenging fun and surprises that that entails, I think we would see a dramatic change both in the attitude of students toward mathematics, and in our conception of what it means to be “good at math.” We are losing so many potentially gifted mathematicians— creative, intelligent people who rightly reject what appears to be a meaningless and sterile subject. They are simply too smart to waste their time on such piffle.

SIMPLICIO: But don’t you think that if math class were made more like art class that a lot of kids just wouldn’t learn anything?

SALVIATI: They’re not learning anything now! Better to not have math classes at all than to do what is currently being done. At least some people might have a chance to discover something beautiful on their own.

SIMPLICIO: So you would remove mathematics from the school curriculum?

SALVIATI: The mathematics has already been removed! The only question is what to do with the vapid, hollow shell that remains. Of course I would prefer to replace it with an active and joyful engagement with mathematical ideas.

Much of the rest of his article is a deconstruction of current mathematics education in public school (what I'd really like to read about is how he'd do it right!) But I am fascinated to consider that mathematics may be a far more complex, artful and exciting world that I'd ever conceived of before and I"m looking forward to exploring in a completely new way...

Can it be four months already?!

I haven't posted much lately (Maddie's been taking over blogging duties!) but I realized on the first day of spring that we've been here four months, through the end of fall and all of winter. Aside from about six weeks of sub-normal (read ridiculously cold) weather from mid-Dec to late January (when I was wondering how many of these winters I'd get through before I started to dread them) the freeze broke and we had lovely late winter weather that was perfectly tolerable.
And now, there are little flowers and strappy leaves poking up here and there in our yard, delightful surprises from past owners that make me jump up and grab a watering can, anxious about these ignored bulbs.
We've got a couple flats of seedlings in the ground (lovely actual dirt again! Not that hard, rocky decomposed granite of Murrieta -- even though it is on the clay-ey side, I'll take it anyday!) in one of our cold frames, and I've got four raspberry plants waiting impatiently for me to decide where their patch will be. Steve's been digging up ground for a strawberry patch, which I'll have to finish up and get planted as well.
Steve's been biking into town in the mornings, for exercise and coffee, which stokes his desire to get an in-town job and stop commuting to Boulder. It will be interesting to see how things develop for him on that front.
He and I got to go on a lovely sunset bike ride during my folks' visit and it made us greedy for more, so we've made arrangements with a friend whose son our kids *love* to play with to trade kids-watching for massage for her! So, more dates on the horizon!
All in all, I think we've made the adjustment here very well. Maddie and Steve are completely gung-ho to be here, and I'm largely happy and enjoying the new places and experiences to enjoy, though I do feel still a bit wistful and transplanted sometimes and missing my folks. I can't say for sure with Harry, but he seems increasingly comfortable here, though when I mentioned our old home and life a few weeks ago, he said he misses it still.
Off to open a box of games that I ordered online, cool imports from Europe that my group of gaming friends reccomended as good family games. They include Settlers of Catan, Metro and The Amazing Labrynth.
When Cindi and the boys come up tomorrow, I'm going to make them play them all with me!

Frosty the snow Monster

We had some snow the other day and in the school yard behind our house we made a huge snowman -- actually a snow Monster!

Just look at how big it is!
I would say about 4 stories tall!



















Actually, this is the real size.























how are you going to throw that!?!?




















Here are some pictures not related to snowmen but are still cute!
oh how i love snow!























ok, I'm stocked up and ready to go!
















Tag! You're it!

RATS

Oh no, there's a rat in the house! There sure is, but it's our pet. Ever since our mice died we have wanted a little pet. We had heard that gerbils were good pets so we went to Petco thinking that we would get a gerbil. At the time, we thought they were solitary but we found out they really aren't. A person at the pet store told us all about them. She said that thay actually do like to be together. She also said that they might not be playful right now because they had just came in and are also young. And then she said that she had 11 rats in her house and one cage for the boys and another for the girls and babies. Now that's a lot of rats! She also showed us some rats that were up for adoption. They were so cute!!! We just had to adopt them!!! So now we have two extremely cute rats named Remy and Emile. Remy is all black. Emile is black-hooded and white. ha ha! I like the names. Here's some pictures of them. Enjoy!

sunny + fang = love

Today my grama and grampa came to our house with their dog Fang, a pure white German shepherd and Sunny's boyfriend too. We moved a month or two ago and ever since Sunny's missed Fang and Fang's missed Sunny. But today (as i told you) they visited us and Fang visited Sunny and of course she was overjoyed! And ever since Fang and Sunny have been playing furiously! Today WOW is down, so I'm whiling away the time blogging. Anyway, Sunny needs a bath again ever since Fang came she's needing baths 24 seven!!!!
(I blame fang) He cant stop licking her!!! And Maggie just wants to sleep (Maggie's our 15-year-old dog in our years. And in dog years 108!!!) I know she's very old but she's doing fine. Astonishing!!!! Well, I can't put up any pics; I'm on one of our laptops. Oh well. Anyway, hope you like this post.

wow!

Today we got into world of warcraft and are having a blast! We are on a few realms we filled up Hydraxis (our main realm) and we're on our way on filling up a couple other relams such as: dragonblight, venture co. and duskwood. And harry has played a big part in doing so. He has the most characters by far. Harry doesn't really seem to like levelling up one character; he likes to make a bunch of characters (human girls mostly) and get them up to level 5 or lower and make more. I on the other hand do enjoy making characters but I mostly like playing my main (a level 16 druid). I don't really like levelling up new characters. I enjoy playing existing characters that can do more spells and so on.

I enjoy druids the most, I would say. They can shapeshift.Starting at level 10 they get bear form, then next is seal form they get at level 16. Seal form makes you swim faster and you don't run out of air when you swim. And at level 20 they can do cat form. I like it the best. Then you can turn invisible (rogues can do it at level 2 if they train.)

That's all the druid forms i know. Our highest level character is a level 21 night elf druid and that's why I don't know any other forms. I also like the druids because they have long-ranged attacks and a spell called entangling roots. It will entangle the enemy in roots and if you get out of its range you can attack it freely with out it attacking you, but if it has long range attacks it can stil hurt you. But it's still not a drain of your mana because it still dose nature damage, a type of attack, but if your target is immune to nature damage it is a drain of mana but as far as i know this would only occur if you were fighting another player and only a night elf at that. So i think druids are pretty darn cool. Harry likes the rogues because they can stealth; that's what druids in cat form can do but rogues get it much sooner.

And there's much more I could type but I want to watch my mom and Harry play. Harry wants to get his rogue a bandit mask (only rogues can wear it). Well, I'm going to stop typing and watch my mom play. Hope you enjoy this post, by!

As all good projects are wont to do...

the remodel has bogged down. After a day of lightning fast demolition and a day of solid reconstruction, Steve came down with a stomach virus, leaving Lincoln to finish up drywall taping on his own. Then I discovered that you can't reliably order bathroom furniture based on your son's name (the manufacturer told me, when I called to check on delivery time, that they actually *didn't* have it in stock, as they told me they did, and it would be six to eight weeks. oy) So, back to Home Depot a few times and down to Lowes in Loveland and we finally have a vanity, counter and medicine cabinet sitting in the garage, along with boxes of tile for floor and walls, whilst Steve wrestles with backerboard and two illustrations for the Times due Friday.
Have we bitten off more than we can chew? Will we finish it in time for Mom and Dad's visit? Stay tuned!