Seeing problems 3-D


Seeing problems 3-D
Originally uploaded by anselm23

I can’t tell you how many times this year I’ve made this exact model. In case you can’t tell, it’s a house. There are variations on this single math problem in every book, in every grade, and I keep trying to show friends and colleagues in other schools and here, that they must be teaching it wrong.

Here’s the problem, and keep in mind that there are lots of variations of this:

“Doris is building a playhouse for her sister. The playhouse is built, and now she is trying to paint it. The paint comes in gallon cans that cover 200 square feet. With the dimensions of the various parts of the house shown in this diagram at right… how many gallons of paint must she buy?”

How do I know they’re teaching it wrong?

Because students from all over keep showing me that they DON’T know how to solve it. The diagrams in the book don’t help them; they are confused by the pictures with their helpful lines and measurements in feet or meters, inches or yards… whatever. The information isn’t enough to help them solve the challenge. They cannot see the problem in three dimensions, on a two dimensional surface.

But I show them one of these quick cardstock models, and they get it right away. They even take my blue Sharpie marker, and figure out how to design the model better, or mark it up in order to solve the problem.

Forced perspective or orthographic projections don’t solve all difficulties. Brunelleschi, the famous Renaissance architect, said that “Errors in the sketch are magnified in the model.”

BUT, if we’re teaching kids to build 3-D card stock models of their math problems, 1) we have introduced a whole new way for students to ‘show their work’ and 2) we’ve allowed them to understand a new dimension to the problem presented. For example, in this quick model, the student and I discovered that Doris’s playhouse wouldn’t need nearly as much paint as we assumed — until the student correctly recognized that the whole back wall of the model was missing… Doris’s playhouse wasn’t complete, when constructed exclusively from what was visible in the diagram in the book.

I DON’T want anyone to walk away from this thinking that I know how to teach math (I don’t, and I shouldn’t), or that I think that my colleagues who teach math are dumb or wrong-headed.  I think they’re really top-notch people… But I also see a place for teaching visual thinking and three-dimensional awareness going un-used, and a general blind spot on both the teaching side and the learning side that is easily filled with a piece of card stock and a pair of scissors.  I’d like to see that change.

Via Flickr:

One of my advisees brought me a math problem she couldn’t see properly. Bad diagram in the text book, perhaps, or just not equipped just yet to understand how the image unfolded into length, width, and height.

So we made a model. And she found that she had a more accurate understanding of the problem than most of her classmates.

Taiji Day 89: I don’t wanna…

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But I do anyway.

There are going to be days like today when you don’t want to. I didn’t even want to get out of bed. I was exhausted at the end of yesterday, and I went to bed at a time much closer to 8pm than anyone should… and I had dreams, though I don’t remember them, and I didn’t awake until 4am or so, cranky and feeling more than slightly cruddy. I don’t want to get up, I don’t want to go to school, I…

I complete the eight pieces of silk. My mindset starts to turn around. I finish the form. And I want breakfast. I do Five Golden Coins, and I… And I’m fine.

The bad mood, the apparent illness setting in, the exhaustion of yesterday, all vanish. There will need to be a good long rest when school finishes up, and lets out. But three forms of chi work, and my mood has improved tremendously. I’m already feeling better and more alert, and more prepared to face the day. It’s what I love about this work — it helps me tell the difference between physically sick days, and psychically-sick days. Today could have been me moaning and feeling sorry for myself. Instead, I’ll be up and about, and a great teacher.

Enjoy the song that I’ll be singing in the shower this morning!

Palace of Memory Page

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I’ve assembled connections to most of the PAlace of Memory stuff I’ve done so far, onto a single page, which has a dedicated link in the page menu bar above… or, you can get to it all using this elegant and well-crafted link.

Inside our Narrative Loop


WIll Richardson, in his recent blog article about the USA Today article about the Khan iceberg, points out that the number and strength of the disruptive forces arrayed against education – both public and private.  I left this comment on his website:

William Lind wrote a series of columns on war titled, appropriately enough, On War, in which he pointed out that wars are won on the moral level long before they’re won on the physical level.

Why bring this up? Because one of the greatest powers a military force can use is to get inside the narrative loop of their opponents, and so thoroughly re-write the story of the war that the some of the opponents’ own allies believe the propaganda; and the story becomes what one side wants it to be, rather than the other.

This is the situation that schools (public and private) are in: we the teachers are being challenged as not tech-savvy enough, not accountable enough, not trained enough, and not agile enough to recreate schools for the 21st century. That’s the narrative presented in the mainstream media, in state and local governments. It’s now common enough that I’ve sat at the dinner table with good friends — who know that I’m a teacher — and yet I still find myself subjected to abusive language calling for, effectively, an end to schools.

Our opponents are inside our narrative, picking it apart from the inside, and they have seized the moral high ground in a lot of ways. Lind’s essays are essential reading for anyone who wants to change the narrative, because first we have to drive our opponents out of our own narrative, and then we have to restructure our own narrative so that they cannot get inside of it.

The fact that we’re having to read essays on war in order to get advice about media strategy is a sign of how far behind in this war for hearts and minds we really are…

I want to develop this theme a little bit farther.  In his essays on the modern-day experience, and how it’s likely to change in the near future, John Michael Greer has recently pointed out Polybius’s model of historythat Greek city-states went through a four-stage cycle which we can characterize as 

  1. Monarchy
  2. Oligarchy
  3. Democracy
  4. Stalemate

He argues… quite successfully, I feel… that rather than being in stage one (where one person alone has taken all power to him/herself) or stage three (where the people have the power), that we are in stage four: that our political and economic institutions are able to continue to hold their existing economic advantages, but not able make significant changes. Nor are reformers able to wring significant concessions or reformation from “the system”.  Nor is any group able to impose its will for a long period of time on the system as a whole — all the other stakeholders fight back against any effort for any one faction to hold or dominate the system.

And it is in this modality that so much effort and energy is being expended at weakening the power of the established educational classes. Without breaking the power of the teachers’ unions, and without dividing the teachers from the students, and the students and their families from the schools, it is impossible to reassign the resources that presently go to schools toward some other effort — like invading Iran, or enriching the .01%, or weakening the political power base which schools and their employees and devotees represent.  Someone… or in fact, as is far more likely, several someones, in several different factions, have decided to go after those resources: that they are but weakly protected, that they are winnable, and that they are worth more than the expected cost to take them.

In other words, disempowering schools and the school teachers that work in them, is part and parcel of the opening maneuvers of a war.

If you’re a local business leader, the tens of thousands of dollars of tax money that go to your local public school system — for bands, for salaries, for football, for textbooks, for whatever — represents tax-breaks your company doesn’t get.   It represents workers who are being mis-trained or even educated to move away from your community.  The school itself is a massive pile of bricks and steel and industrial grade spaces that can be rented, borrowed or even stolen.  The workers in those schools are overpaid intellectuals WITH THREE MONTHS OFF each year.

Suddenly, along comes Khan Academy.  Along comes several billion dollars in venture capital funding, and hundreds of new content-delivery ventures.  Along comes YouTube, Wikipedia, and a host of other resources and websites that provide ‘free educational programming.’ Of course, you need to pay for an internet connection, pay for a computer, pay for access… but hey, captive audience! The kids, they have to go to school, right? THeir parents have to figure out how to occupy their kids learning math facts and when the Battle of Gettysburg was, and they’ll have to pay someone to help them educate their children.

Why should that money go to the state? Especially when those teachers have done such a bad job of it? Especially when the schools flunk all the standardized tests? Especially when the state has to figure out how to serve poor people’s kids along with rich people’s kids?

Understand. I’m not saying that I believe these things to be true.  I went to private schools for most of my education, and I teach now in a private school.  But I’m not sure that I agree that private education in the US substantially outperformed public school until standardized testing and NCLB substantially hamstrung the public schools, by forcing them to concentrate on the testing process rather than providing a well-rounded education.

Wars don’t break out in a moment of relative stability. They break out in a time of incredible disruption.  The American Civil War broke out at a moment when a completely unknown candidate took the oath of office for the presidency, and the Confederacy thought it could win autonomy with the support of the European industrial powers.  World War I? European aristocratic ideals meeting the new industrial technologies of railroad, machine gun and howitzer.  World War II? The fossil fuels meeting  meeting the old industrial technologies of the previous war.  

I can’t say I know too many people who really, really like schools.  Parents tolerate them because it provides cheap-ish babysitting services.  Politicians tolerate them because people leave town and destroy tax bases if they’re bad; but they’re expensive to maintain, and they tie up both capital and revenue in order to run them.  Teachers like them for the paycheck, but apparently enough teachers feel that teaching gets in the way of their lives, that they don’t do enough of it, or at least not enough to suit some colleagues or parents (See comments on this entry from a long time ago, “I don’t know any incompetent teachers“).

Schools, rightly or wrongly, are seen as treasure houses. From the Harvards and Yales to the lowliest public school in Montana, they are concentrations of wealth and influence and power.  There are people who would like that power, or to share in that power, or to break that power, and they will dismantle that power structure if given a chance.

It may be too late to prevent them from doing so. 

The scoffers may tell me that I am being needlessly apocalyptic, and they may be right. But all of this has happened before, and all of it will happen again.  The notion that the schools cannot be dismantled wholesale or piecemeal should not be dismissed as mere fantasy.  

End of Year

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There’s a big push beginning today at school, which will last through graduation on June 8th, and then into end-of-year meetings for teachers the following week. There’s also writing end-of-year progress reports, and suchlike.  Expect low-level blogging, unless I see a vision during a taiji session of the way education could be remade, if only everyone would listen to me.

That said, I’ve embarked on two rather complex projects this week.  It’s often the case that the stress and complexity of end-of-school-year issues triggers my creativity in new ways. It’s not always HELPFUL to have these new currents or energies arise at this time of year, but they do.

There are two new jars in my spagyrics cabinet, and a new file or three on my computer… so far, so good.

Taiji Day 88: Heart

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It’s already quite hot and humid in my apartment, and it’s now 5:20 in the morning, and I’m already sweating. All the exercise people tell us that we need to sweat, and elevate our heart rate, for a certain amount of time before the benefits of exercise kick in. If that’s the case, more than half of my taiji workouts do nothing (Form, five golden coins, eight pieces of silk this morning… and I was reminded of another one that I have knowledge of in a dream [thanks, segmented sleep!] last night.) But I feel like more and more of them are getting to this point of being “light to moderate exercise” — that is, I’m getting a workout, not just moving around in some poses.

Taiji Day 87: two sleeps


I did Taiji: eight pieces of silk, the form, and five golden coins. Not bad, not good. This is getting to be normal.

The other side of it is bad sleep habits from almost two decades, which I’m trying to fix. About two months or so ago, I realized that both Jason Miller’s course and Frater RO’s course were urging students to reconnect with their dreams. I wanted this, badly. When I’m healthy and getting enough sleep I have amazing dreams. But how much was enough? How do I get that much?

The trick seems to be (for me) is using the concept of segmented sleep, as found in a lot of traditional societies. Sleep dysfunction is a relatively new phenomena: people in pre-industrial societies got up in the middle of the night (paywall, now, alas). A Roger Ekirch has done some work on this, and published a book called AT DAY’S CLOSE, about this segmented-sleep concept… I have to admit, I haven’t read the book, but got a review of the idea from the New York Times, and from the BBC.

In any case, I’ve been trying to bed down between 9:00 and 9:30 on weeknights. I wake about 3:00 from a good but vague dream which I’m able to recall in some detail and write down. Then I sleep again from 3:30 to 5:00 am. That’s 7.5 hours of sleep or so, which is a lot better than my old five hours. The dreams in the second sleeping session are clearer, more defined, sharper, more vivid and more easily recalled and written down. It seems to be a good system.

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